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For the purposes of this question, we're going to ignore the rules, found in either the Player's Handbook (pg. 187) or Xanathar's Guide to Everything (pg. 134), that allow a character to learn a tool proficiency as part of their adventure downtime. For our purposes, we assume that these methods of gaining tool proficiencies are not available to us.

So given that constraint and the constraints below, what is the maximum quantity of tool proficiencies a character can learn as part of their normal progression?

The constraints:

  • Only officially published 5th Edition sourcebooks
  • No Unearthed Arcana, however tempting it might be to include the newly released Artificer revision there is an officially released version of the Artificer, and that is valid.
  • Temporary effects, i.e. tool proficiencies gained from from spells should only be included if the character in question is capable of applying/maintaining said effects themselves.
  • No proficiencies gained from magic items
  • No use of downtime activities to learn new proficiencies (PHB pg. 187 or Xanathar's pg. 134)
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12 Tool Proficiencies, 15 if you count musical instruments, 16 if you also include gaming sets.

Let's take this piece at a time.

Race

Human with the Prodigy Feat, Envoy Warforged, Rock Gnome, or Dwarf. Nets +1 Tool Proficiencies.

Background

Any House Agent background (Eberron), except Kundarak, Medani, or Tharashk (Because those grant Thieves Tools, which is a waste of a slot) Nets +2 Tool Proficiencies

Feats

Skilled, obviously. +3 Tool Proficiencies. That's the only feat that grants Tool Proficiencies, and you cannot take it more than once.

Classes

Bard: 1 level IF you are counting musical instruments. They get 3 Musical Instrument proficiencies at level 1. This must be your First Class, or you only get 1 musical instrument

Rogue: 3 Levels, Mastermind. +3 Tool Proficiencies (Thieves' Tools, Disguise Kit, Forgery Kit) and +1 gaming set

Cleric: 1 level, Forge Domain. +1 Tool Proficiency (Smith's Tools) OR Knowledge Domain (Channel Divinity to gain temporary access to Proficiency with any 1 Tool)

Monk: 3 levels, Drunken Master. +1 Tool Proficiency (Brewer's supplies) OR Kensei (Calligrapher or Painter's Tools)

Fighter: 3 levels, Battlemaster. +1 Tool Proficiency (Your Choice)

Spells

There are no spells that grant temporary proficiency. The only magical effect that lets you 'fake' proficiency is found in the Knowledge Domain Cleric, which only lasts for 10 minutes (not long enough for a major project), but it's a choice between that one, and a Forge Domain cleric (which grants a permanent proficiency). So, do you want short term flexibility, or a full proficiency?

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    \$\begingroup\$ Could probably add the Vedalken (Guildmaster's Guide to Ravnica, pg. 21) to the race list, since they also gain a single Tool proficiency as part of their Racial features. \$\endgroup\$ – Xirema Mar 4 '19 at 20:41
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    \$\begingroup\$ It might be helpful to cite the sourcebook for things from outside the PHB/core books (e.g. Eberron content). \$\endgroup\$ – V2Blast Mar 5 '19 at 4:51
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The question asks not to include the Artificer from Unearthed Arcana. Now that there is an official Artificer class from Eberron: Rising from the Last War, I will present this answer that includes the official Artificer.

(Note that much of my answer is borrowed from the other answers, I've just included a few levels of Artificer.)

Through levelling only, you can have a total of 19 tool proficiencies!

(This assumes musical instruments and gaming sets count as tool proficiencies; if not, then we're down to 15 tool proficiencies with this build; replacing the starting level in Bard with a starting level in Druid brings this up to 16 thanks to Herbalism Kit proficiency; alternatively, if you first level is in Artificer, you would also have one more choice of tool proficiency, so 16 is the maximum either way without musical instruments or gaming sets.)

Race

  • Human, to gain access to the Prodigy feat;
  • or Dwarf, for Mason's Tool or Brewer's Supplies (avoid Smith's Tools);
  • or Vedalken (Guildmaster's Guide to Ravnica, pg. 21), for a tool proficiency (thanks to this comment for pointing that out);

(I'm avoiding Rock Gnome because we want Tinker's Tools from multiclassing into Artificer later; also note that by not picking Human, you miss out on one more from the Dragonmark below).

Dragonmark

  • If we use Dragonmarks from Eberron: Rising from the Last War (p. 45), you can pick the Variant Human: Mark of Making, which grants proficiency with a set of artisan's tools (via the Maker's Gift racial trait). This still counts as Human, so is still eligible for the Prodigy feat.

Background

  • House Agent (from Eberron: Rising from the Last War, p. 53), which gives you two proficiencies:
    • choosing House Ghallanda for Brewer's Supplies and Cook's Utensils, which works if we avoid Brewer's Supplies via Dwarf and avoid the Drunken Master for our Monk subclass;
    • choosing House Jorasco for Alchemist's Supplies and Herbalism Kit, which works if we choose Artillerist for our Artificer subclass instead of Alchemist (although Alchemist subclass lets us choose a different tool proficiency if we already have Alchemist's Supplies, so actually it doesn't matter which Artificer subclass we pick);
    • choosing House Thuranni for a Musical Instrument and Poisoner's Kit, which works if we are accepting Musical Instruments in our total;
    • choosing House Deneith, Lyrandar, Orien or Vadalis, which works if we are happy to include Vehicle proficiencies in our total;

Classes

  • Bard 1 (taking our 1st level in Bard gets us three Musical Instrument proficiencies)
  • Cleric 1 (Forge Domain, to gain Smith's Tools)
  • Monk 3 (Drunken Master or Kensei; the former gives you Brewer's Supplies, so don't pick this and get Brewer's Supplies via Dwarf, and the latter gives you either Calligraphy Supplies or Painter's Supplies)
  • Rogue 3 (Mastermind, which gets you Thieves' Tools from multiclassing into Rogue, then Disguise Kit, Forgery Kit, and a Gaming Set proficiency from Mastermind archetype)
  • Fighter 3 (Battlemaster, which gives you one set of Artisan's Tools)
  • Artificer 3 (we get Tinker's Tools from multiclassing into Artificer, also Thieves' Tools but we already have that via Rogue; we also get one more depending on our archetype, so Alchemist or Artillerist would be our best choice as that gets us Alchemist's Supplies or Woodcarver's Tools, which we don't have yet; if we did pick Battle Smith, we would already have Smith's Tools but it lets us pick a different tool proficiency if we already have Smith's Tools, so actually we can go with any Artificer subclass...)

Feats

  • Skilled (from this we can gain proficiency with any 3 sets of tools)
  • (If we chose Human as our race, the Prodigy feat will give us an extra proficiency, but that's already accounted for under Races above)

Total

So that's:

  • 1 from race (picking "Human" assumes you take the Prodigy feat at some point)
  • 1 from Dragonmark "Mark of Making" (assuming race is Human)
  • 2 from background
  • 3 from the Skilled feat
  • 3 from Bard
  • 1 from Forge Domain Cleric
  • 1 from Drunken Master/Kensei Monk
  • 4 from Mastermind Rogue
  • 1 from Battlemaster Fighter
  • 2 from Artificer (any subclass)

For a total of 19 tool proficiencies.

Final Build:
A "Mark of Making" Variant Human Bard 1/Cleric 1/Monk 3/Rogue 3/Fighter 3/Artificer 3, with the Prodigy and Skilled feats by taking two of those classes that we already have three levels in one level further.

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    \$\begingroup\$ I went ahead and updated the question just to make it clear that the official version of the Artificer is, indeed, valid for this question. \$\endgroup\$ – Xirema Jul 10 '20 at 15:42
  • \$\begingroup\$ @Xirema Cool, thanks for that. \$\endgroup\$ – NathanS Jul 10 '20 at 15:43
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The maximum number of tool proficiencies is 17
If musical instruments do not count, this maximum drops by 2.
If gaming sets do not count, this maximum drops by 1.

The Character

Connor is a human with the Dragonmark of Making (in Wayfinder's Guide to Eberron) and these classes:

  • Bard 1 ------------------------- (for musical instruments)
  • Cleric (Forge from XGtE1) 1 --------------- (for smith's tools)
  • Monk 4 ------------------------ (for 1 set of artisan tools or 1 musical instrument, 1 other set of tools, and the Prodigy feat from XGtE)
  • Rogue (Mastermind from XGtE) 4 ------ (for thieves' tools, disguise kit, forgery kit, 1 gaming set, and the Skilled feat)
  • Fighter (Battlemaster) 3 ---- (for 1 set of artisan tools)

1: XGtE refers to Xanathar's Guide to Everything

The Tools

Note: If musical instruments count, bard must be chosen at level 1. If musical instruments hold less than half the weight of other tool proficiencies, then choose Druid instead for herbalism kit and multiclass into bard later for one musical instrument

Your Monastic Tradition can either be Drunken Master (for brewer's supplies) or Kensei (for painter's supplies or calligrapher's supplies), both from XGtE.

Your Roguish Archetype must be Mastermind unless gaming sets do not count in which case you can also choose Assassin (which replaces forgery kit with poisoner's kit)

Dragonmark of Making grants an artisan tool proficiency of your choice.

The Prodigy feat grants any tool proficiency of your choice, while the Skilled feat grants you 3 of your choice.

A customized background can grant any two proficiencies in tools according to the [Player's Handbook or Basic Rules.

To customize a background, you can replace one feature with any other one, choose any two skills, and choose a total of two tool proficiencies or languages from the sample backgrounds.

Final tally:

  • 3 musical instruments of your choice -or- herbalism kit and 1 musical instrument of your choice
  • smith's tools
  • disguise kit
  • thieves' tools
  • forgery kit and 1 gaming set of your choice -or- poisoner's kit
  • 3 sets of artisan tools of your choice -or- 2 sets of artisan tools and 1 musical instrument of your choice
  • 6 of any tool of your choice

Extra Stuff

  • As a Mark of Making human, you can add a d4 to every ability check with a set of artisan tools.
  • As a Rogue you can choose to gain Expertise in your thieves' tools.

Temporary Proficiencies

If musical instruments do not count, an alternative build can acquire a different proficiency than herbalism kit, albeit temporarily, using the spell magic jar on an NPC in Waterdeep (spoilers):

Thorvin Twinbeard in Waterdeep - Dragon Heist

If you forgo the level in Cleric and only get 3 levels in Monk and Rogue, you can replace Bard 1 with Wizard 11 to get magic jar which will both replace the smith's tools you would've gotten from cleric as well as grant you mason's tools. (You then use the wizard's Ability Score Increases to get the feats required).

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