After reading through all of the complaints about rogues losing combat utility in exchange for better skill usage, I decided to see how well one could optimise a specific skill (and how much of a factor being a rogue would be in that). As it turns out, you can auto-succeed "Nearly impossible" (DC 25) tasks.
I decided to choose Bluff for this sheer hilarity of what you can do with a sufficiently high check. See Order of the Stick
for some examples.
The basic setup is fairly simple:
Race: Stout Halfling (for +1 Cha and a 2/day reroll from Lucky)
Class: Rogue (for Skill Mastery)
Scheme: Trickster (for Bluff, also seems appropriate)
Background: Largely Irrelevant, but Jester, Sage, or Thief all work.
Speciality: Skills (but with Skill focus taken twice rather than Superior Skill Training, but this isn't really necessary)
Select Bluff and Persuade for Superior Skill Training and Skill Focus, and Bluff for Skill Supremacy
Put 4 skill points into Bluff and 2 into Persuade (or 3/3 if you prefer)
Start with at least 17 Cha and increase it every four levels, for a total of 20. Dump the other two points wherever you like.
Purchase/acquire the following items: Pale Green Prism Ioun Stone (+1 to checks) and a Potion of Heroism (+2 to checks)
Befriend a Cleric and have them cast Guidance (+1 to a checks using a specified ability) on you
Putting it all together:
+5 for 20 Cha
+7 for Skill Bonus
+2 for Superior Skill Training
+1 for Ioun Stone
+2 for Potion of Heroism
+1 for Guidance
+ 3d10 pick highest for skill mastery
Total: +19 to +28, average +26
But that's not all!
As Bluff is typically a contest (vs Wis/Sense Motive), your minimum roll is 10 (arguably this also makes a natural 1 not an auto-fail). So the least you can roll is 29. So much for DC25 (or equivalent opposition to the check) being near impossible.
But that's still not all!
You also get advantage on Bluff rolls, and can reroll one of the dice again. This gives an average of 15.74 (notably, 3d20 pick highest gives an average of 15.5, so the ability to take 10 doesn't seem to be doing much). Clearly you can pretty much convince anyone of anything with an average of 41.75.
So what do you do with this? Whatever you want! Go convince the king that he is actually a pretender and you are the rightful king. Convince the High Priest of Pelor that you are the Avatar of Pelor and it's His will that all the gold in the temple be put in the cart(s) outside. If you crit (with three rolls, you have about 13.75% chance of at least one 20 [14.25% according to anydice.com]), and at least one d10 shows a maximum, you have a check of 48. Nothing short of a god is going to doubt you.
Alternative uses for this would be to spec sneak and sleight of hand instead (starting at 17 Dex and being a Lightfoot Halfling instead), and steal the kings robes *While he is wearing them.* Or you could steal the enemy leader's armour, if you want to actually contribute to a combat.
No sane DM would let you do this, but there aren't really any questionable interpretations of rules here. The only two areas where interpretation is needed are whether Superior Skill Training can improve skill bonus beyond the +7 cap (I've assumed it can, because no cap is mentioned in the document containing the feat, let alone in the feat description itself), and and whether magic items are available in your campaign. Reduce the numbers stated by 2 if the former is deemed false (but you free up a feat), and reduce them by 3 if the second does not apply. Outside of those two issues, all they could throw at you would be Rule 0 or Wheaton's Law (and they would be perfectly justified to do so).
As a side note, almost all of this could be done by a fighter. You could still get an automatic 28, or 24 without items/spells. It's a bit less extreme, but still completely broken. And in exchange for that, you get all of the combat benefits of being a fighter (but you can't disable traps, or use more than 5 skills [7 if you give up Lucky and switch to being an Elf] - and only 1 of them can be broken). Still, being able to get a check of 48 in a supposedly bounded-accuracy system is rather disconcerting.