Yes, it may be available
What is possible
- There are Incantations, variant rules for 3.5e that perfectly fit into the concept of E6, allowing you to invest a lot of effort and get a limited version of a high-level spell effect. Your GM has to give you access to those.
- With successful checks you can cast spells from 4+ level Scrolls, if GM lets you access said scrolls.
- Versatile Spellcaster feat allows Clerics and Druids to cast 4th level spells by sacrificing two 3d level spell slots. Clerics also have to make an extra 1-hour long spell preparation session this day, read below the "How is it possible?" section for details.
- Artificers are cheaters who can cast high-level spells out of the box starting at character level 5.
- With some word stretching and extreme investments (well, extreme for the wealth level of E6 characters) it is possible to cast 5th level spells, and perhaps even spells of higher levels.
It might break your game
The point of E6 is actually excluding high-level spells from the game. Because they break the game, because it is hard to do anything against (some of the) 4+ level spells without other 4+ level spells to counter, which forms that huge gap between casters and non-casters, making normal 3.5e and Pathfinder games very "unfair" starting at 7th level, where prepared spellcasters get their first 4th level spells normally. Quoting KRyan's answer about purposes of E6:
Lower-level spells are undoubtedly powerful, but for the most part, they can be handled or mitigated by a lot of characters—characters with good Reflex and Balance can ignore grease, characters with capabilities that benefit fighting blind are much less affected by glitterdust, fly is a relatively short-duration spell, and so on.
This becomes less true with 4th-level spells. Solid fog allows you absolutely no recourse unless you can teleport. Enervation is extremely difficult to defend against (unless you have death ward, which is also 4th-level), and absolutely devastating in effect. Divine power allows a cleric to basically turn into a barbarian on top of his spellcasting. Dimension door and freedom of movement are the beginnings of spellcasters’ “get out of jail free” cards. Polymorph is just basically cheating. And divination, lesser planar ally, and scrying can completely change the nature of game.
And, of course, you might look at that and say “well yeah, 4th-level spells are better than 3rd-level spells; that’s how things are supposed to be.” But the real problem is that non-spellcasting classes do not get any such power bump. What 4th-level spells offer are a series of attacks that are exceedingly difficult to defend against without 4th-level spells, as well as a series of defenses that are exceedingly difficult to penetrate without 4th-level spells. The rogue gaining another +1d6 sneak attack damage, the barbarian gaining DR 1/– (the fighter and the ranger getting literally nothing), those are just small incremental improvements, not a whole new league of power the way 4th level spells are.
And it gets worse, much worse, from there.
But it doesn't mean that it will ruin your game: if you carefully choose which spells are allowed and which are not, it may add an interesting flavour to your game, making your spellcasters feel really epic without spoiling fun of other players.
How is it possible?
There is a guide by Forrestfire on GiantITP.com called "High Priests and Archmages: Plot Magic in E6". This guide introduces two terms for characters built in a way to allow using 4 and 5 level spells in E6 rules: "High Priest" for a cleric, and "Archmage" for an arcane spellcaster, hence the guide's name: it is generally full of fluff text which might be fun, but I will try to develop the ideas mentioned in this guide, concentrating more on the RAW aspect of achieving the needed results.
As noted in the guide:
The artificer can, out of the box, begin to scribe scrolls and make wands of 4th-level spells off of any spell list, starting at 5th level. This means that many options, such as reincarnation or animate dead, are available to them at a cost of gold, experience, and time.
Cleric, the High Priest
According to that guide, the following trick may be performed.
Getting level 4 spells
Use the Versatile Spellcaster feat. The Prerequisite is "Ability to spontaneously cast spells", which all Clerics qualify for, being able to cast either Inflict or Cure spells spontaneusly. Feat text:
You can use two spell slots of the same level to cast a spell you know that is one level higher. For example, a sorcerer with this feat can expend two 2nd-level spell slots to cast any 3rd-level spell he knows.
The intended feature is most likely burning your unneeded low-level spell slots to cast spells you actually have access to, but we are breaking the game, so it's not important, right? Let's check if a 6th level Cleric may actually know any 4th level spells...
Quote from Cleric class description from d20srd.org:
A cleric may prepare and cast any spell on the cleric spell list, provided that they can cast spells of that level.
That is almost exactly what we need. Not specified that the spells are known, but certainly available. So, we can most likely combine two level 3 spell slots and can cast one level 4 spell, and hence, according to the quote above, we can deploy it after preparing it.
However, we also know that our Cleric needs one hour to fill their empty spell slots with spells:
Each cleric must choose a time at which he must spend 1 hour each day in quiet contemplation or supplication to regain his daily allotment of spells
So, you need one preparation session to get your spell slots back. Then you combine your regained 3d level slots and prepare a 4th level spell in another meditation session, you are allowed to have as many of those as you need.
Getting level 5 spells
This gets considerably harder.
You need to convert four level 3 spell slots in two level 4 spells: your normal two 3d level slots you have at level 6, your Domain 3d level slot, your bonus slot for Ability score (Wisdom 16+ needed). Cast those two spells, you need them used, not ready.
Versatile Spellcaster feat doesn't convert 3d level spells into 4th level spell slots, it only allows you to cast and prepare level 4 spells. How do we get those 4th level spell slots? We need the Siphon spell! RAW, it allows you to spend 5 charges from a Wand containing at least 5 charges of X level spells "to replace spells you have already cast". Notice this very important line:
If you prepare arcane spells, you can regain any one spell of the affected level that you have already cast that day; if you cast spells spontaneously, you can regain a spell slot of the appropriate level.
You still remember that all of the Clerics can cast spontaneously? So, after casting our two 4th level spells, we can spend a total of 10 charges from a 4th level Wand to get two 4th level spell slots. The Siphon spell itself you have to get from a scroll or a wand, and you will have to roll an appropriate Use Magic Device check, as Clerics don't normally cast spells from Wizard/Sorcerer list. 10 charges of a 4th level spell Wand cost 10/50*21 000 GP=4200 GP, a charge of Siphon costs 11 250 GP*1/50=225 GP.
It actually turned out that this part of the loop has limits, as a 6th level Cleric doesn't normally have level 4 spell slots, they have nothing to regain. Some other means of converting 4th level spells into 4th level spell slots needs to be found. You still have the ability to cast 2 4th level spells as a level 6 caster, which is already powerful. Sticking with 4th level spells is also free in terms of money. Or you need to stretch the wording of "regain" a bit.
- Use Versatile Spellcaster again to convert those two 4th level slots to cast and prepare one 5th level spell from Cleric spell list, you need to have your third preparation session.
Notice, again, that you need:
- The Versatile Spellcaster feat from Races of Dragon, p. 102: the book has to be allowed by your GM, and if you were playing Pathfinder, using this feat is actually homebrew (I couldn't find the respective Pathfinder feat)
- All of your 3d level spell slots. That is a thing to consider.
- Play with the wording of which spells do you know and which do you not know.
- The Siphon spell from Complete Scoundrel, p. 102. Again, in 3.5e the book has to be allowed by GM before you use it, and in Pathfinder that requires porting 3.5e material.
- 10 charges from a 4th level spell wand. So your GM has to actually have already given you 4th level magic. Perhaps in may be crafter by the Artificer.
- Access to the Siphon spell as Cleric, even if it is allowed: you need a good Use Magic Device score, and in a E6 campaign a poor roll can spoil everything.
- Time for your three spell preparation sessions per day.
You don't need points 4 to 6 if you are satisfied with casting two 4th level spells (no 5th level).
Your GM has plenty of buttons to disallow that if having no desire to see 5th level spells in his campaign. But even if you happen to persuade them successfully, remember, again, that it was actually the point of E6 to throw high-level magic away. Such a trick is a lot of effort, and it may make for an interesting story hook -- perhaps a whole session dedicated to reviving your Paladin who died at the end of previous session by Raise Dead! Need to deal with a negative level, though, so you can cast Restoration the next day by burning two of your 3d level slots for one 4th level spell.
As soon as Druids also cast spells spontaneously (Summon Nature's Ally) and hence also qualify for the Versatile Spellcaster feat, they may also perform the same abuse as the Cleric to cast a 5th level Druid spell. Or simply stop at 4th level and cast Reincarnate (though this might be "a bit" risky, it is 5 times cheaper in terms of materials, and you don't need 10 charges of a 4th level Wand, the Siphon spell and stretching word meanings).
Beguiler, Dread Necromancer, Warmage
Pretty same abuses, but let's just check that they will have access to 4th and 5th level spells that way:
When a dread necromancer gains access to a new level of spells, she automatically knows all the spells for that level given on the dread necromancer's spell list
When a warmage gains access to a new level of spells, he automatically knows all the spells for that level listed on the warmage's spell list. Essentially, his spell list is the same as his spells known list. Warmages also have the option of adding to their existing spell list through their advanced learning ability as they increase in level (see below).
When you gain access to a new level of spells, you automatically know all the spells for that level on the beguiler's spell list. You can cast any spell you know without preparing it ahead of time. Essentially, your spell list is the same as your spells known list. You also have the option of adding to your existing spell list through your advanced learning class feature (see below) as you increase in level.
The Sorcerer (and other spontaneous casters)
To cast a 4th level spell, you have to learn it in the first place, which doesn't happen normally until you get to 8th level. But you can buy a Page of Spell Knowledge (costing 25 000 GP) and learn one spell, then, with the help of the same Versatile Spellcaster you will be able to cast it 2 times per day (by burning all of your 3d level Spell Slots. Well, 25 000 GP is a lot for E6 characters... Why not just play T1 guys if you are looking for a way to break the system?
You can use the advanced slots for metamagic, though, not paying for the Page of Spell Knowledge.
Again, you are breaking the purpose of E6
If you are OK with that way of playing, why don't you just go further and modify E6 to stop at 8th level or simply play the game normally, advancing without E6 rules?
There is no resurrection available to PCs in E6 under normal circumstances, that makes life a valuable thing. There is no teleportation available either -- makes sense for castle walls to exist, and so for ship, post-houses etc. No Atonement means that it is very important for your Paladin to adhere to their guidelines -- etc.
Each 4+ level spell can make your game less meaningless by excluding one more problem from it -- and D&D is actually a game about solving problems by making decisions.
Reasons to write this
- Because link-based answers aren't good, and it seems nice to have a full explanation here, on RPG.SE, instead of relying on the page on giantitp.com
- To understand the thing better myself. This is crazy.
- Because I have found out that the potential of this trick is even greater than listed in the original guide (you can actually cast more spells this way than just your Domain spells).
- Because the original guide has, in my opinion, too many "fluff" words. I am focusing more on the RAW aspect.
- Because it was found out that casting level 5 spells this way actually doesn't work, or requires crazy level of word stretching.
As soon as E6 is compatible with both 3.5e and Pathfinder, I have some Pathfinder entries in this answer.