How far do you think a single level 20 character could get through an ap, run completely raw? What class/archetype might work better than others?
Technically, the character would have the strength of a level 16 full party.
In practice they will be far stronger in certain things, while lacking the versatility required to face some less conventional challenges.
If I was to choose a pure class to play such an adventure, I would probably go with either a rogue or a caster. Wizard has got the most tools to solve any kind of problem, given enough time, but it also lacks combat healing for when the real punches start flying.
With archetypes/multiclassing, I would use some combination of rogue, wizard and/or druid.
All the way?
Part of that would depend on the AP of course, but not just its creatures, but the winning circumstances. I'm not familiar with all (or even most) of the endgame situations, but final bosses that can be thwarted (rather than directly defeated) should be doable, though with significant risk of course.
Rogue
Not simply for the skills/skill feats to cover extra roles, but Blank Slate at later levels would allow them to bypass many enemies. And that's not the only ability that works great w/o other party members to worry about, though yes, it'd make Sneak Attack more difficult to apply.
Creatures w/ Regeneration or Fast Healing would be an issue, but with time the others could be whittled down through sniping.
The get an MCD that opens up magic options, likely a prepared caster so they can build to suit their newest obstacle.
For a more straightforward approach, Monk.
High AC + kiting/bypassing via speed + self-healing in combat and more
Shield Ally Champion's a runner-up, though its difficulty vs. flying creatures w/ ranged attacks would be problematic. They could be the ones kited. That'd require a shift in build which might undermine the reason to choose them.
I'd avoid anything with a mediocre Will save!
A Sorcerer might be fun, blasting 3 Focus Spell AoE's every 10 minutes, though I doubt they could push through unless they had perfect information (and time to Retrain their spells as needed).
A Fighter w/ Boundless Reprisals, a Reach weapon, and another way to get Reach (so you get AoOs vs. Large melee creatures too), might cut through the first 16 levels the fastest! Add Sudden Leap to cover lots of space and Whirlwind Attack (and a way to get Haste to get into position).
That'd be a nightmare to enemies.
Speaking of which Scare to Death might get most any 20th level PC through most enemies. If one built around that by taking a class that works well vs. enemies immune to fear/emotion/death effects (like a martial), they could get to the last AP pretty fine.
ETA: Scare to Death would suggest a class that can afford to invest in Charisma, which not all can do.