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I only have the PHB and DMG. I'm the usual DM.

My friends are more interested in using Cantrips than developing high level spells. More for fluff than for combat variety, to be sure.

I'm sure RAW doesn't permit casters to collect more cantrips than listed without expending Feats (ex: Magical Initiate or Spell Sniper) but I'm wondering if the Unearthed Arcana or other books have variant rules that make it possible for PC's to accumulate more Cantrips.

Homebrewing-wise one idea I had was to permanently substitute a PC's spell slot for each additional Cantrip acquired.

But I was hoping that there were rules out there guiding a high-cantrip count build. I'd prefer if answers not focus on multi-classing, which is an obvious path, yes, but has little to do with the direction of my question.

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There are no WotC-released rules for this that you aren't already (per your Question) aware of

I've gone through all of the Unearthed Arcana and all the rulebooks and no such "get more cantrips instead of spells" variant rule exists.

The ways released by Wizards of the Coast to acquire more cantrips than your class and class features would naturally give you are Multiclassing, Feats, or starting play as a race that gives you a Cantrip such as a High Elf or Tiefling.

There are variant rules to allow you to cast more low level spells (see DMG288), but those aren't what you are looking for.

Homebrew Territory

As DM, it is entirely in your wheelhouse to invent rules to allow players to 'learn' new Cantrips. There are a few different ways I have seen this conducted. Unfortunately, I have not played with any of these rulesets, so I can't speak to how well they work in-game. But, hopefully, this will give you some ideas if you want to take the plunge into homebrewing up rules.

(Disclaimer: I don't remember where I saw each of these first, so I can't give you the original source)

Spell Preparing Casters can Prepare Cantrips

The gist of this rule is that it expands the rules for preparing spells to include Cantrips. Thus, for Clerics and Druids, they may prepare up to their 'known' number of cantrips each day, choosing any cantrips from their spell list. The Wizard works much the same way, but must have these Cantrips in their spellbook (they 'add' a new cantrip automatically any time their 'known cantrips' count increases--and can add new cantrips to their spellbook from spellbooks and scrolls as if they were 1st level spells).

Downtime to Learn Cantrips

Using the 'Research' (or referencing the 'creating a magic item' rules if you want it to be harder) Downtime rules from UA and, now, Xanathar's Guide I have seen homebrew rules that allow any caster who can cast Cantrips to use Downtime to learn a new Cantrip.

You can learn extra Cantrips as 1st-level spells

This rule essentially means that any Cantrip that you don't know as a Cantrip can be learned and treated like a 1st level spell. It isn't a spell you can cast by rote at this point, and thus consumes a Spell Slot, but it does give you access to Cantrips you wouldn't have otherwise. Paired with the Spell Points option on DMG288, this could let your players cast them a LOT as they get higher level.

I've seen a small variation on this where it suggests that every time a player casts a Cantrip as a 1st-level spell, the DM should roll a d100 or something...and on a 00, the player has 'mastered' the cantrip, and adds it to their list of Known Cantrips

Your own Suggestion...

If you want to permit a player to 'give up' a learned spell in exchange for another cantrip, then go for it (i.e. a Sorcerer should learn a new spell, but you allow them to learn a new cantrip instead). This is actually rendering the character a bit underpowered because spells are, naturally, more potent than Cantrips.

Summary

As mentioned, there are no WotC-released rules allowing you to gain more cantrips outside of Multiclassing, Race Selection, Feats, or the Cantrips (and at-will Warlock Evocations) granted by your class and class features. But, if you want to take the jump into Homebrew-land...hopefully these ideas I've seen around the net will help.

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The most common ways to get cantrips are, as you mentioned, from your class and from feats. There are some other options available. Not all options are suitable for all campaigns or are compatible with preexisting characters. I've roughly ordered the options from most player control (leveling options) to least (DM fiat).

Arcane Domain Clerics: From the Sword Coast Adventurer's Guide, these fellows gain two Wizard cantrips in addition to their usual complement of Cleric cantrips.

Warlock: Warlocks get some straight-up cantrips. They also get Invocations. Like cantrips, Invocations offer at-will magical abilities. However, these tend to run more powerful than cantrips, particularly the level-restricted items. Warlocks may also choose the Pact of the Tome, giving them an additional three cantrips.

Multi-classing: When you multiclass into a full caster (Bard, Cleric, etc.), you get a few new cantrips. If your only concern is maximizing the number of cantrips, then being very multiclassed into the full casters (Bard, Cleric, Druid, Warlock, Wizard, Sorcerer) is the way to go. The down sides include losing out on other higher level abilities of the class you started in as well as the roleplaying baggage of having chosen another class.

Race: Certainly not something you usually change mid campaign, there are a few races that offer specific cantrips (Tieflings) or your choice (High Elf).

Other Options: If your characters aren't interested in casting higher level spells, perhaps they'd be interested in a half caster, such as the Paladin or Ranger, or a one-third caster, such as the Fighter or Rogue. Note that Paladins and Rangers don't get cantrips, but do get other quasi-magical abilities of interest.

Magic Items: Since it's not an option at character creation or level up, this is DM territory. There are a few items that allow casting a spell at will, such as the Hat of Disguise (disguise self) and Ring of Jumping (jump, self-only).

Quest Rewards: The DM may choose to offer an unusual reward for a quest. For example, completing a particularly arduous quest for Pelor, God of the Sun, may result in him granting you the light cantrip by deific fiat.

Stretching: You could attempt to cast wish (or get it cast for you), and simply wish to know more cantrips. This really opens you to DM retribution, with basically no limits. A wizard Wishing to know the cantrip druidcraft could find himself turned into a level 1 druid. Messing with the fabric of reality is not advised.

Stretching Further: If your goal is "cast a ton of weaker spells" and you simply expressed that as "have more cantrips", then I recommend you take a look at DMG 288 for its Variant: Spell Points rule. This is effectively more of a mana pool than the typical Vancian magic system. That is, you could use your magic for a whole bunch of level 1 spells and simply not cast anything of a higher level. A level 20 character could cast 66 level 1 spells in a day using this rule.

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    \$\begingroup\$ -1, I think this was posted after a key edit was made to the original post, which is seeking answers that do NOT focus on multi-classing. I would recommend an edit to this answer to address the querent's intended focus. Please tag me after a change is made and I can rescind the downvote and possible upvote. \$\endgroup\$ – Pyrotechnical Dec 29 '17 at 15:25
  • \$\begingroup\$ @Pyrotechnical I've reworked the answer a bit to focus less on multiclassing and more on options the players have vs. the dm having for gaining cantrips. \$\endgroup\$ – Joel Harmon Dec 29 '17 at 21:22
  • \$\begingroup\$ +1 I think that improves the answer a good bit and also covers most of the possibilities for the querent. You only touched on magic items, but there's at least one gives the user access to cantrips (Staff of the Magi includes Light and Mage Hand, but I'm not sure if others do as well). \$\endgroup\$ – Pyrotechnical Dec 29 '17 at 21:30
  • \$\begingroup\$ @Pyrotechnical I thought it was pretty clear that the Magic Items list was intended to be exemplary, not exhaustive. I just did a quick thumb through to find at-will spells, not specifically cantrips. If I interpreted the question too liberally, I can certainly edit in the Staff. I also skipped Artifacts, which can confer cantrips if attuned. \$\endgroup\$ – Joel Harmon Dec 29 '17 at 21:52
  • \$\begingroup\$ I only mention it because the querent is seeking answers related to cantrips, not spells. The information you provided isn't wrong, but as a subsection in your answer I think it could be improved to directly address what is being sought. \$\endgroup\$ – Pyrotechnical Jan 2 '18 at 15:24
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I have gone through all the unearthed arcana up to 11/28/2017. Here are ways to gain cantrips:

  • Dragonmarks: these are marks on skin and grant you a cantrip and use of some spells as you increase in level. A dragonmark is taken as a feat.
  • City domain clerics gain the on/off cantrip at first level. (On/off is a transmutation cantrip.)
  • Warlock patron 'ghost in the machine' grants the on/off cantrip.
  • Warlock patrons 'undying light' and 'celestial' grant the sacred flame and light cantrips.
  • Abyssal tiefling race variant grants one random cantrip at 1st level. (Later UA give a list of particular cantrips for different fiend lords.)
  • Grave domain turns the casting time of the spare the dying cantrip into a bonus action.
  • Artificier subclass gunsmith grants the mending cantrip at 1st level.
  • Wood Elf Magic feat can be taken by wood elves and allows you to pick one druid cantrip.
  • Eladrin has a racial feature that allows casting of one of friends, chill touch, minor illusion, or fire bolt; the choice of which can be repeated after each rest.
  • Githyanki and githzerai can cast mage hand as a racial feature.

So nothing of the sort that you are asking appears to exist: All ways of gaining new cantrips require either multiclassing, or choosing a given race, or picking up some feats.

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RAW

Various class options get you a fair number to start with.

  • Warlock gets a base of 4. Taking Pact of the Tome will get you three more on top of that, from any spell list. Additionally, many invocations are at-will.

  • Cleric gets a base of 5, and taking arcane domain gives you two more wizard cantrips.

  • Sorceror gets 6 straight-up.

  • Bard gets up to 4, and College of Lore adds two more on top of that from any class.

  • Wizard gets a base of 5, but the Illusion School gets Minor Illusion as a free cantrip.

  • Druid is the real loser here, with just 4, but they make up for it in other ways.

Additionally, the feats you've mentioned, but there are also races that give cantrips

  • High Elf gets a wizard cantrip of choice
  • Tiefling gets the Thaumaturgy cantrip

There are also some magic items that give access to at-will spellcasting of various sorts. Hat of Disguise, Decanter of Endless Water, Ring of Invisibility, and Ring of Telekinesis stand out as providing such in particular.

Beyond that, RAW has blessings and boons that you can be granted. "extra cantrips" would be pretty low-key as such things go, really.

Homebrew

First, let's talk balance. Magic Initiate is a feat that offers you two cantrips and a 1/day casting of a first-level spell. It's also a feat your players aren't interested in. Thus, we can be sure that a cantrip is worth less than half a feat to them, and you can hand them out in any situation where you might otherwise hand out feat-fragments, without fear of imbalance. In particular, Tomes offer permanent statgain. You could quite reasonably whip up a few tomes of cantrip knowledge, and probably drop them by one rarity from the standard stat+2 tome.

It's also worth noting that the more cantrips you have, the less useful each additional cantrip is - partially because you'll normally grab the best cantrips first, and partially because once you have a decent number to play with you might start running into a bit of overlap in terms of the problems they can solve. This is much less the case for cross-class cantrips, though, and in particular offering cross-class cantrips modified to use the base class casting stats could open up specific combos. (Cha-based shillelagh on a pact of the blade warlock with polearm master, for example, which is normally only available by also taking 6 levels of bard).

The bigger issues, though, have nothing to do with crunch balance. The first is option paralysis. Eventually, having too many choices may slow things down as everyone tries to remember everything they can do and figure out which to apply to the situation at hand. That depends on your players. More to the point is character uniqueness. If one guy in the party has the ability to light the campfire by looking at it funny, that's a cool thing that he can do. If everyone has that power, then it's that much less cool, and there's a limit to the number of interesting cantrips to go around. Eventually, a party full of cantrip-hunters will start doubling up.

I also note that the lore bard gets to select either cantrips or additional spells known. That's in keeping with the idea that extra cantrips (after the first few) are a boost to versatility but not to power.

So, in a game where everyone is a spellcaster, offering a homebrew that would let them cash in spells known/prepared in exchange for cantrips is probably pretty reasonable, as long as you limit the number of additional cantrips they can get. It won't break your game balance, certainly. I wouldn't suggest it, though. I'd suggest having extra cantrips be the sort of thing they can earn as rewards from time to time. You've already said they want them, and having your casters be driven by the quest for additional magical knowledge and power is entirely in keeping with the genre.

In a game where only some of the people are spellcasters, you can still do that, but you might want to keep an eye out so that the spellcasters with their cantrips for every occasion aren't taking too much of the spotlight from the less magical characters.

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