Wizard
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:
Shugenja
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.
Best Answer
It's hard to tell
Throughout the editions of D&D, Mordenkainen has been associated with all sorts of schools of magic. Firstly it is important to note that Mordenkainen is an expert at almost all forms of arcane magic. The Epic Level Handbook of D&D 3e says:
It is clear that Mordenkainen has vast enough knowledge that he could rival any specialist in their field. That being said, we can look at various publications to get an idea of what types of spells Mordenkainen values/focuses on.
1st Edition
In the 1st Edition era, Mordenkainen showed up in many books, but the one I want to focus on is Mordenkainen's Fantastic Adventure, an adventure module where Mordenkainen is presented as a pre-generated character, a level 12 magic-user:
As can be seen, even at Mordenkainen's earliest appearance, he had access to all spells he was capable of casting, but if we break down the spells he had readily available based on this list, we get this breakdown:
\begin{array}{c|c} \text{School} & \text{Count} \\\hline \text{Alteration} & 7.5 \\\hline \text{Evocation} & 5.5 \\\hline \text{Abjuration} & 3 \\\hline \text{Enchantment} & 2 \\\hline \text{Conjuration} & 1 \\\hline \text{Divination} & 1 \\\hline \text{Illusion} & 0 \\\hline \text{Necromantic} & 0 \end{array}
This suggests that even though Mordenkainen could use all spells, his go-to list at the time focused primarily on Alteration (which later became Transmutation) and secondly Evocation.
3rd edition
A much older and more powerful Mordenkainen made an appearance in the 3rd edition supplement Epic Level Handbook where he once again knows just about every spell. In this case, however, we don't get a basic list of spells that Mordenkainen typically prepares. That being said, this Mordenkainen has epic spells which have to be chosen rather than prepared, of which he has chosen:
One of these is a conjuration spell, and three are transmutation spells, once again lending the idea that Mordenkainen favors transmutation spells.
5th edition
By the time 5th edition rolls around, Mordenkainen has lost a bit of power after certain events that I wont spoil since they consider multiple adventure modules. However, Mordenkainen is basically reduced to a generic archmage. The spell list we are given for him in his first appearance in 5e is as follows:
Ignoring cantrips (since those can't be prepared), we get:
\begin{array}{c|c} \text{School} & \text{Count} \\\hline \text{Abjuration} & 5 \\\hline \text{Conjuration} & 4 \\\hline \text{Evocation} & 4 \\\hline \text{Divination} & 3 \\\hline \text{Transmutation} & 3 \\\hline \text{Illusion} & 1 \\\hline \text{Enchantment} & 0 \\\hline \text{Necromancy} & 0 \end{array}
This distribution is quite a bit more evenly spread than the 1st edition list, but with a favor for Abjuration spells. Granted, the events that led to this weaker version of Mordenkainen also had dramatic effects on his memory and personality, so it is possible that his focus may have changed as a result.
Invented Spells
Another way to think about Mordenkainen's specialization is to look at the spells he has invented. These spells include (I've excluded the prefix "Mordenkainen's" for brevity):
\begin{array}{c|c} \text{Spell} & \text{School} \\\hline \it{buzzing} \text{ } \it{bee} & \text{Conjuration} \\\hline \it{capable} \text{ } \it{caravel} & \text{Conjuration} \\\hline \it{celerity} & \text{Alteration} \\\hline \it{defense} \text{ } \it{against} \text{ } \it{lycanthropes} & \text{Abjuration} \\\hline \it{defense}\text{ } \it{against}\text{ } \it{nonmagical}\text{ } \it{reptiles}\text{ } \it{and}\text{ } \it{amphibians} & \text{Abjuration} \\\hline \it{disjunction} & \text{Abjuration} \\\hline \it{electric}\text{ } \it{arc} & \text{Evocation} \\\hline \it{encompassing}\text{ } \it{vision} & \text{Alteration}\\\hline \it{faithful}\text{ } \it{hound} & \text{Conjuration} \\\hline \it{faithful}\text{ } \it{phantom}\text{ } \it{defenders} & \text{Summoning}\\\hline \it{faithful}\text{ } \it{phantom}\text{ } \it{guardian}& \text{Summoning}\\\hline \it{faithful}\text{ } \it{phantom}\text{ } \it{shield} \text{-} \it{maidens} & \text{Summoning}\\\hline \it{force}\text{ } \it{missiles} & \text{Evocation} \\\hline \it{involuntary}\text{ } \it{wizardry} & \text{Enchantment/Necromancy}\\\hline \it{lucubration} & \text{Alteration} \\\hline \it{magnificent}\text{ } \it{mansion} & \text{Conjuration} \\\hline \it{penultimate}\text{ } \it{cognition} & \text{Alteration} \\\hline \it{private}\text{ } \it{sanctum} & \text{Abjuration} \\\hline \it{protection}\text{ } \it{from}\text{ } \it{avians} & \text{Abjuration} \\\hline \it{protection}\text{ } \it{from}\text{ } \it{insects}\text{ } \it{and}\text{ } \it{arachnids} & \text{Abjuration} \\\hline \it{protection}\text{ } \it{from}\text{ } \it{slime} & \text{Abjuration} \\\hline \it{sword} & \text{Evocation} \\\hline \it{trusted}\text{ } \it{bloodhound} & \text{Conjuration} \\\hline \end{array}
This gives us the following distribution:
\begin{array}{c|c} \text{School} & \text{Count} \\\hline \text{Conjuration/Summoning} & 8 \\\hline \text{Abjuration} & 7 \\\hline \text{Alteration} & 4 \\\hline \text{Evocation} & 3 \\\hline \text{Enchantment} & 0.5 \\\hline \text{Necromancy} & 0.5 \\\hline \text{Divination} & 0 \\\hline \text{Illusion} & 0 \end{array}
As you can see, among the spells Mordenkainen invented, he definitely showed favor towards conjuration, but I'd say more so abjuration. I say this because many of the conjuration spells are abjuration-like in function, and the most powerful spell Mordenkainen invented, Mordenkainen's disjunction, is an abjuration spell.
Conclusion
Overall, it is clear that, over the years, Mordenkainen has primarily focused on the Schools of Transmutation and Abjuration when it comes to spells prepared and invented. I would wager that these two schools are the ones Mordenkainen is most familiar with and has the deepest aptitude for.
All that being said, because Mordenkainen has been such a wide-ranging spellcaster in the past, it is unclear whether this favoritism is actually a demonstration of higher skill, or just a manifestation of spells he prefers. After all, someone as wizened as he (having known just about every spell a wizard can), isn't really lacking in any school.
Another thing of note is that Mordenkainen seems to disfavor necromancy and illusion magic across just about all of his appearances. This doesn't necessarily mean he is unskilled with this type of magic (as we know he has many spells of this ilk in his repertoire even if unprepared). I just found it interesting to see.