(the following is a bit of a train-of-thought working through some mechanics and balance, feel free to skip down to the class table
To preserve balance, there have been different approaches: Magic users "forgetting" their spells after casting them, slower leveling progression, etc. Warlocks can use many of their spells without limitation, though they are of lower power than wizard or sorcerer of equal level could cast.
But most magic users of fiction, be they like Gandalf, or Harry Potter, or from somewhere else, rarely work that way. They can use magic (within their knowledge) almost without limit. Some fictions might have magic that has a fleeting drain or exhaustion that becomes apparent as the caster reaches their limit, but usually that's about it. Can we do the same thing in D&D?
Psionic classes in the current and past few versions of D&D (and some sorcerer variants) have used power points as a replacement for spell slots, which gives some additional versatility, but generally has relied on a once-per-day or long rest reset. If power points refreshed more quickly, can we get some semblance of balance?
Let's start with the spell's cost. If we make cantrips cost 1 power point (pp), then we can let everything else costs 2pp for each level of the spell (so a 3rd-level fireball would be 6pp). Now, just looking at cantrips, then, if the caster regenerates 1pp per turn, we've got our status quo -- the caster can cast cantrips every turn, all day long.
If we stick with the 1pp/turn regeneration rate, let's look at how big the caster's power point pool should be. It needs to be large enough that they can caster their highest level spell, but we want it smaller than, say, what a psion gets since it will regenerate much faster. It probably needs to be a bit bigger than the cost of the highest spell, but only by a little bit. Caster level plus one or two or three or ability bonus might be good. Let's say it's caster level plus three, which is probably the same as caster level plus ability bonus for most low-level characters.
A first level caster would have 4pp. And that the 1pp regen happens at the start of the turn, and only if the caster is below their full pool amount, They could cast a three first level spells (assuming 1 per turn), and then be stuck with cantrips unless they went a round without casting. Over an "adventuring day" of 6 to 8 encounters, that's a fair bit more powerful than a normal 1st level wizard or sorcerer, who could only cast 2 first level spells total.
At level 5, the caster would have 8pp. A 3rd-level fireball would cost 6pp, and then the caster would have 2 rounds of 1st-level spells, and then be down to cantrips. But the caster could have some more interesting choices to make: Maybe they plan on using 2nd level spells for the first two rounds of the battle, instead, or just use a cantrip the first round or two before deciding to use a 3rd or 2nd level spell. This is getting closer to being in line with a regular caster's 9 1st-or-higher spells per day, but still likely above the average.
From the GM's point of view, this is quick-regen style caster class actually makes magic users little less of a wild card. Regular casters can choose to blow through all their highest available spells at once, making the current encounter trivial but making a possible later encounter that much harder. While that can be an interesting decision, it is often one that is made without meaningful information, as rarely do the characters have knowledge of exactly how many encounters they might face before having a chance to rest. With a quick-regen power point pool, the caster will have their top spells available every encounter, but with a limit on casting too many of their high level spells in succession.
If your game contains both sections of single encounters separated by days of rest, as well as dungeon-crawls filled with many back-to-back battles, then I think you would see casters being on a much more even keel compared to martial types.
That said, what we've got now means that a 5th-level caster could still cast a fireball a minimum of once every encounter. If 6 to 8 is the expected number of encounters in a day, that's way above the 2 times a regular wizard could do it.
Let's lower the pool threshold. If we make the pool cap the caster level then the caster can't ever cast their highest level spells. The could, then, have a once-per short rest ability that gives them a few extra power points and raises their pool cap temporarily (alternatively, it could reduce the cost of casting a spell, same difference). With an average 2 short rests per day (plus the initial rest going into that day), the caster could now hit their max-level spell just 3 times a day. This is getting closer to what we want. Our level 6 caster, though, can still cast fireball every encounter, so let's tweak a few more numbers.
We want the highest level available spells to be only available with a 'boost" ability. If we stick with the double-the-spell's level for the power point cost, then we want a 1pp limit at levels 1, and it being 1 less than caster level beyond first level. The boost ability can give 2 or more power points (and is allowed to exceed the regular power point cap), so only when the boost is triggered will the caster get their highest level spells.
So, we've got a caster class that can use their highest available level of spell once each rest. They'd be able to use spells below their highest level more often, but unless a fight runs long and the caster take rounds out to not cast anything, they caster will still be limited to a few of those bigger spells.
One drawback is that by giving cantrips a cost, a caster needs to not use them to rebuild their power point pool. One of the design goals of 5e's cantrips was that caster could use them for reliable, stylistically-appropriate damage (instead of the situation where the magic user spends a lot of their time shooting a crossbow). We could drop cantrips to having 0 cost. A caster, once low on power points, would be able to then effectively alternate between casting cantrips and 1st-level spells, or stick to cantrips for a few rounds if they wanted to rebuild their pool.
One last consideration is out-of-combat spells. If you feel that they need to be limited, you could assign certain spells a drain that reduces the power point pool cap by a point or two until a long rest is completed. This would stop them from being able to be cast too many times a day.
Here's our final (for now) version:
|Class level||Power points||Maximum spell level|
|Spell level||Power points|
Spell casting: You can cast a spell up to the maximum level listed, by paying the power points for it out of your pool. If your start your turn below your pool's maximum, you regenerate 1 point.
Boost: You can cast a spell spending 2 power points less than you normally would. This ability resests on a short or long rest.
Some other fun features might include stuff like reducing the cost of spells of a certain school by a point, or the caster being able to increase their regen rate occasionally.
I'll think about some more features, and hopefully try to playtest this a bit, it should be a class that better captures the feel of fictional magic users, but doesn't overshadow other classes (at least in theory).