It is well known there are seven prohibitable schools of magic, of which the Focused Specialists can prohibit three. I'm unsure of how these stack (in other words, can you ban the same school twice?), but the Incantrix and Red Wizard prestige classes both require an extra school to be prohibited. With prestige classes such as these in mind, is it possible to create a Wizard that has every prohibitable school prohibited? And if so, does this cause any issues when the character tries to level up (e.g. being unable to gain bonus spells)?
It turns out that topquark’s fine answer finds a couple more options:
changeling wizards from Races of Eberron are another wizard variant with 3 prohibited schools (but probably isn’t strictly compatible with focused specialist, so you can’t go to 4 that way).
wizards of high sorcery from Dragonlance Campaign Setting are another prestige class that prohibits a school of magic, but has restrictions on it.
That answer also points out that you cannot prohibit your own specialization, since the rules for prohibiting a school say you must choose “other schools” for that, and all the relevant prestige classes refer to those rules.
The changeling wizard ends up being a red herring, though: it probably doesn’t quality for focused specialist, so it’s no better than focused specialist is, and its dual specialization is problematic for the wizard of high sorcery. But if we ignore the changeling option, we actually do better.
A focused specialist/wizard of high sorcery/incantatrix/red wizard would prohibit 6 schools, leaving Divination and their specialty un-prohibited, but also un-prohibitable, since Divination never can be and you also can’t prohibit your specialization. The order of classes here matters, because both wizard of high sorcery and incantatrix have limitations on what they can prohibit, so I order them from most limiting to least limiting, to make sure we are able to use all of them. There is, as topquark mentions, still an issue with both the wizard of high sorcery and the red wizard using the same name, “enhanced specialization,” for their class features, but the rules for handling this situation are rather vague—I think most DMs would have them stack.
Note, of course, that even though our specialist cannot prohibit his own speciality, that speciality is still arguably “prohibitable” in the general sense, since it can be prohibited by someone, just not him. In order to get around that, you would need to be a diviner, but then you would also need to find a way to prohibit two more spell schools: even focused diviners only prohibit two spell schools, plus we would need a way to prohibit the school that would otherwise be the specialized school.
The long and short of it is that this is a heavily qualified “yes,” but still falls well short of the likely goal.
But that’s the wizard, which, it turns out, is not the only class that prohibits some spells. And if you thought that was a heavily qualified answer, well, get a load of this:
The Complete Divine shugenja does as well—not by school, but by elemental categories that are rather similar to schools (since they go well beyond simple elemental descriptors). Almost all shugenja spells are associated with one of Air, Earth, Fire, or Water, and all shugenja must choose to specialize in one of these and prohibit the opposite element. So, for example, a shugenja could specialize in Earth spells—and would therefore prohibit Air spells.
Meanwhile, a raptoran who can cast summon monster IV or summon nature’s ally IV could take the skypledged prestige class from Races of the Wild—which requires prohibiting all spells with the earth, fire, and/or water descriptors.
So if a raptoran shugenja of an order specializing in Earth were to somehow get summon monster IV or summon nature’s ally IV into their spells known—neither is a shugenja spell—they would have Air shugenja spells prohibited as well as all spells with the earth, fire, and/or water descriptors. Shugenja do have a few spells designated All, but these are reasonably considered “unprohibitable” since no shugenja order does so.
However, there are a lot of caveats here:
Strictly speaking, this has nothing to do with spell schools, which the question did specify. Though shugenja elements are basically an alternate set of schools. Complete Divine calls them orders.
There are spells that a shugenja considers Earth, Fire, or Water that do not actually have the earth, fire, or water descriptors (for obvious examples, electricity falls under Fire and ice falls under Water; more esoteric examples like Air getting illusions or Water getting healing are also there). The skypledged prohibition specifies earth, fire, and/or water descriptors, and these spells don’t have them.
The skypledged prohibition is vastly laxer than the shugenja or wizard prohibitions: a skypledged can still cast these spells. They just lose their non-spellcasting skypledged class features—a big blow, no doubt, but it’s still well short of “you simply cannot.”
The skypledged gets access to a divine spellpool—the ability to get some spells off of the cleric or druid spell lists. Since the cleric and druid spell lists don’t have the same elemental categories that the shugenja list does, it’s unclear if the shugenja prohibition would even apply to it. Our shugenja here may well be able to pull Air or [air] spells from the spellpool, and use them despite our devotion to Earth. If nothing else, all of the spells with no elemental descriptor and that aren’t on the shugenja spell list are presumably fair game, which leaves a whole lot of spells available.
This character is fundamentally absurd: this shugenja has a religious devotion to Earth while simultaneously being pledged to the sky and air elementals in their battle against the other elements, Earth included? It seems like one or the other those those vows would be considered broken somewhere along the line here. The rules for each class don’t actually mention any such restriction, of course, but one’s DM may well do so.
So this character has kind of prohibited nearly all of their spells—after all, if the prohibition is voluntary anyway, and the shugenja considers all of the spells listed under Earth, Fire, and Water to be spells of that element even if they don’t have the descriptor, they may well feel beholden to avoid those spells as well. And then they could mentally categorize all of the cleric or druid spells under All—or under some element they won’t use.
But as I said, a lot of caveats and asterisks on that.
There are two more ways to add prohibited schools, besides the ones you've already listed:
- Changeling Wizard racial substitution class (source: Races of Eberron) gets 2 specialist schools and 3 prohibited schools at level 1. Your specialist schools must be Illusion and Transmutation.
- Wizard of High Sorcery (source: Dragonlance Campaign Setting) is a PrC with a drawback similar to Incantatrix and Red Wizard.
Unfortunately, while that would add up to 7, there are several reasons why you can't combine all of them onto one character:
First, I don't think you can prohibit your own specialist school(s), due to the wording of the Wizard class feature:
The wizard must choose whether to specialize and, if she does so, choose her specialty at 1st level. At this time, she must also give up two other schools of magic, which become her prohibited schools.
... and all of the other class features that mention prohibited schools probably inherit that restriction. So if you're a Changeling Wizard, then you can't prohibit Illusion and Transmutation by any means.
Second, Wizard of High Sorcery has some restrictions about which specialist schools can be paired with which prohibited schools, and those are onerous enough there there simply is no valid choice for the 4th prohibited school.
Third, Wizard of High Sorcery's class feature has not only the same effect but also the same name as Red Wizard's class feature, "Enhanced Specialization". Which usually means they don't stack.
Fourth, it's unclear whether Changeling Wizard is compatible with Focused Specialist. Changeling "replaces the standard wizard's specialization option", and it's unclear whether what you get in return also counts as "a specialist wizard" i.e. the prereq for Focused Specialist.
And finally, I'm mixing 3 different campaign settings. YMMV about whether that's an issue.
As a completely different approach to the whole problem, you can prohibit all of the spells without involving specialist or prohibited schools per se.
A Planar Wizard (source: Planar Handbook) at level 10 permanently adds any one alignment descriptor to all of your spells, as well as gaining some bonuses vs opposite aligned targets. This doesn't have to match your character's alignment at all.
Silver Pyromancer (source: Eberron Five Nations) is an arcane PrC with similar code of conduct to a Paladin.
So if you're a LG Planar[evil] Wizard / Silver Pyromancer, then all of your spells are evil acts, so they're all prohibited by your code of conduct.
(Obviously this requires some perverse in-character decisions, but no more perverse than the goal of being a spellcaster who can't cast any spells.)