Bottom line up front: Skald. Plenty of other classes are more powerful, even without all-18s compared to an all-18 skald, but I don’t think any other class is quite so happy to see all-18s.
To begin with, spellcasters are powerful independently of their MAD; being SAD is just one of many things they have going for them. So even with all 18s, even though they really don’t need it, spellcasters are still the most powerful. I think you grasped this, but just to be clear and for completion’s sake.
Moreover, note that rolling all 18s does not solve MAD problems – for a truly MAD class, at mid-to-high levels the impossibility of magically improving all of those ability scores at the rate a SAD class would, means that SAD is still far superior. This is particularly true in Pathfinder, as opposed to 3.5, because Paizo decided to shaft MAD classes even harder by moving all enhancement bonuses to physical ability scores into items that take the same item slot (belt), and all enhancement bonuses to mental ability scores into items that take another item slot (headband). If you need more than one physical score, or more than one mental score, you are going to be paying more than double what a SAD class would.
So this is less “what is the best class with all-18s” (the answer to that is the same as it would be with only one score in the 16-18 range), and more “what class does all-18s let me play that would otherwise suffer too much MAD?” Furthermore, this answer is going to restrict itself to discussing low-to-low-mid levels (say, up to about 9-10 at most), since the benefit of all-18s wanes as you get into higher levels.
Cleric and Druid
Just, to mention them: they’re super-powerful, they like Strength and Constitution pretty well, in addition to Wisdom, and the cleric certainly doesn’t mind Charisma. They kind of qualify, except that they can get by on just Wisdom if they have to.
The monk unchained gets a lot of nice things that came from qinggong monk. I haven’t seen one in play yet, but it should be comparable for this purpose.
Monk certainly is MAD, but that’s just one of its many failings. Even with all-18s, the core monk struggles mightily.
That said, the qinggong monk is a mini-spellcaster, which still has the monk’s need for Strength, Dexterity, and Constitution. Having 18s in all of those stats definitely allows a qinggong monk to make better use of his or her physical options, while retaining spellcasting ability. Sadly, the lack of any good way to work the magus’s Spell Combat or Spellstrike abilities limits the ability to fuse the two neatly.
Which leads us nicely into the magus itself. While definitely a few steps down from a fullcaster, the magus is reasonably fun and as a physical+magical class, makes good use of Strength (or Dexterity, if Weapon Finesse), Constitution, and Intelligence. Obviously, Dexterity and Wisdom have their uses for everyone, so having a high score in those is nice. Without Perception as a class skill, Wisdom 18 just about counteracts the loss of the +3 class skill bonus, which helps avoid surprise. The high Charisma falls a bit flat, but magi have Use Magic Device in-class, so you effectively start with a +7 or +8 to that compared to a warrior who dumped Charisma and did not have it.
With caveat that I have only read the class, never played it or seen it played.
Again following the trend of the physical+magical classes, the bloodrager has the neat ability to automatically apply a spell to himself when he starts a bloodrage. That’s nifty. The class also wants substantial Strength, Constitution, and Charisma, which puts a lot of those 18s to use. Again, Dexterity and Wisdom are some of the better scores for their general use without specific class features, and Intelligence is nice enough for the skill points.
That said, the spells are few and weak, and the ability to auto-cast them on rage comes late (11th for 2nd-level-and-lower spells, 20th for any spell). This is a fairly solid idea for a class but the limits placed on it are just far too tight. A bard progression rather than paladin progression, with the auto-casting coming online from the beginning and not limiting the level of spell used, is what this class needed.
I bet you didn’t see this coming. Doesn’t have the unique bloodrage auto-cast thing that the bloodrager does, but the skald is basically a bard with better armor and weapon proficiencies, and a more combative bent to its music.
Better, the skald really would like to put Strength, Dexterity, Constitution, Intelligence, and Charisma to good use. Going first is particularly important for the skald, because you want your music bonuses to be in effect ASAP, the skald can do melee quite well, and the spells and music depend on Charisma.
I have not gotten an opportunity to play with most of the unchained classes or hybrid classes. Hybrid rules, in particular, make likely targets here, since classes that combine magic with martial ability are likely to risk MADness.
Also, the Path of War material from third-party publisher Dreamscarred Press is excellent, and each of those classes needs a minimum of three ability scores (Strength-or-Dexterity, Constitution, and one mental score). The design there is tighter than Pathfinder’s typical fare, and in particular they do take some steps to limit the problems of MADness, but they still benefit quite a bit from high scores all around.