First, spoilers in this thread as to how GOOs impact the game, how to prepare, beat them, etc.
Second, no Kingsport GOOs on the list since I don't own that expansions
Third, subjective point of view
Combinations that I've played with (adding new expansions to the previous combo, never taking anything out):
AH + DH: 15
AH + DH + DP: 26
AH + DH + DP + KiY + BG: 24
Each GOO has been involved 6 times, with various Heralds. Record against each listed as win-draw-loss (draw being victory by final combat). For the 7th rotation, I'll probably use those Heralds that haven't yet been in a game with the GOO I draw. Just need to figure out what to do with those GOOs that have already used all the Heralds.
#12: Azathoth 6-0-0 (BG, DH, DP, KiY) JtWT
Poor Azathoth, doom track is too long and has a bare minimum of game effect. At least final combat is HARD! No doubt I'll get games where I'll just draw weird combos that will wake up Az, but during an average game, you should have no trouble putting Azathoth back to sleep.
#11: Shudde M'ell 5-1-0 (BG, DH, DP, KiY)
Yes, that's right, Shudde. Again, pretty much negligible game effect. For one, you can go whole games (yes, plural) without seeing a Cthonian. For another, if a monster surge occurs, only the Newspaper is a location that I don't like getting rubbled. Double-resistances can be a hindrance, but the only time I had to fight him, I had 6 turns to punch him (I think normally, 4 investigator team beats GOO in around 4-5 rounds of FC).
#9 (tie): Cthulhu 5-0-1 (BG, DH, KiY) & Tsathoggua 5-0-1 (BG, DH, DP, KiY)
Yuppers, the Big C. After you're done setting up, game impact is definately lacking, plus a 13 doom track is on the long side. Fight is HARD, thanks to the stupid regeneration, but you shouldn't have trouble close/sealing him (accidents can of course happen, C woke from too many gates open, thanks to drawing 2 "a gate and a monster appear" during a turn).
Tsathoggua, again 13 doom track. Oddly enough, Tsat has a couple of game effects that should make things harder, but so far haven't been affected too much. For one, closing everything but Shops, Hospital and Asylum. Mostly not a big deal, but Science Building not available for 2 Clue Tokens isn't nice. Church and Ma's, not a big deal for games around here. Secondly, Urban Environments get tossed without coming into play. Normally, doesn't necessarily matter, but if everyone's favourite Mystic Environment "No One Can Help You Now" hits the table, it should stick around longer than normally. Fight is really nasty, getting Clue chucked at the Start of Battle negates using them. Then, pure monster killers are as useless as pure gate closers. Everybody's gotta multi-task (not a problem here, we don't do set roles for investigators).
#8: Shub-Niggurath 4-1-1 (BG, DP, KiY)
+1 toughness usually means faster spending. With the Black Goat expansion (esp. the Herald) Endless Dark Young can be a real pain. Unless you're really short on magical weapons, final combat should be challenging (thanks to the -5), but not impossible. Meeting Shub-Niggurath through an OW encounter on turn 3, with a fully loaded (Elder Sign, 3+ Clues) investigator with 0 monster trophies, pricess!
#7: Hastur 3-3-0 (BG, DP, KiY)
8 Clues to seal, that's definately not nice. Doom track is nice and long though, but those 8 Clues takes time to gather. Anything that can give you extra Clues (Old Journal, King in Yellow, Press Pass, etc.) and/or Elder Signs are even more welcome than before. Final Combat, well, depends on what number the Terror Track is on. The worst I've had to face was -4 mod. Since 99.9% of the shopping around here takes place at Curiositie, magical weapons aren't hard to come by. Thanks to the Luck check and 2 Sanity damage, spellcasters tend to be less than useful in this (when using Spells that is).
#6: Nyarlathotep 3-2-1 (BG, DP, KiY)
Pattern here, see it? A couple of 12 doomers and 13+ doomers all statistically easier ones. From here, it's all either 10-11 doomers or 12 doomers that have severe impact on the game. Nyarly, 11 doom track, so less time to waste. Luckily, while collecting Clues (and sealing gates), you're also doing what you need to win or survive in the Final Combat. Nyarly doesn't really impact the game (Endless Cultists mean more fodder), you just have less room to hit a bad patch during the game. -4 and Magical Resistance shouldn't really be a problem, unless you only brought spellcasters into the game.
#5: Yig 1-4-1 (BG, DP, KiY)
Yigster, Yigglypuff. Call him what you want, he's a speed-freak. 10 doom track, doom token for LiTaS and killing Cultists (doesn't help when the first 5 of 6 monsters drawn in a game are Cultists either). Not hard to avoid defeat against, but things can happen, it's Arkham after all. Curse isn't fun, but you should get rid of it hopefully in a few turns. Of course, there are those people who get Blessed just before Yig wakes up... To win, I've found gotta hope for gate openings in the major hot spots, then seal them, hoping for seals to snuff Mythos cards or then get monster surges (and not play with the Black Goat Herald). Marie Lambeau can buy that extra round you need.
#3 (tie): Abhoth 2-2-2 (BG, DH, DP, KiY) & Ithaqua 2-2-2 (BG, DP, KiY)
Abhoth is a sneaky one. 11 doom track and looks like minimal game impact. But Abhoth hits you during set-up (and then during game), not having Cultists removes the standard easy fodder, making every monster draw that much more nasty. Like with the other 11 doomers, gotta hope for a relatively smooth ride. Final Combat is all about making sure people have things to discard. As you can see, sometimes there aren't enough items to discard and losses happen.
Ithaqua, my early nemesis. 5th try was my first close/seal against Ith and that came with no less than with the Black Goat Herald! On the first 4 tries, always seemed to get some totally bizarre Mythos card draws. Ithaqua's Cultists love clogging the streets and with +2 toughness, you're looking at some hard buggers. Still recall my first game where Joe D took 3 damage while trying to kill a Cultist in the streets while rolling 12 dice per attack. Final Combat? Argh! Ithaqua loves nothing better than to eat all your usable items. Doesn't help when you're rolling 11 dice while Blessed, failing 10 of those. Biggest hurdle is the start of battle, after that, not too much to worry about (if you have weapons left of course). If you're Carolyn Fern with Heal, you can pretty much outlast Ithaqua with your bare hands.
#2 Yog Sothoth 2-1-3 (BG, DH, KiY)
2 successes to close? LiTaS equals devoured? Ouch! 12 doom track sure, but you should pretty much expect to try close rolls more than 1 turn or spend some Clues (or use Mandy re-roll). Then again, I still recall my Cursed Jacqueline Fine sealing a gate to Yuggoth (3 dice rolled) without Clues! Of the devourings that have taken place during games, Yog probably has caused majority of them. Of late, LiTaS has been a rarer occurrence (of course, I wrote that bit before today's Yog + Black Goat game where Gloria Goldberg drew 2 OW cards that both would cause her to go LiTaS!). Final Combat is nasty. Will checks and need gate trophies? My only draw came after optimal initial possessions: 2x Tommy Gun, .45, .38. During the game, picked up another .38 and Deputy's Revolver. Oh yeah, and Joe Diamond met John Legrasse at Unvisited, started with +1 Will, got +1 Will and +1 Fight during the game! So even with Fight maxed, he had Will 4 and if he ever failed, 1 Clue token would earn him 4 more dice for the Will check. Oh yeah, and I had done Sealing the Beast's Power during the game. Generally though, Yog's a hard customer. And don't let Dunwich Horror be wake when Yog wakes (if using DH Herald).
#1: Glaaki 1-3-2 (BG, DP, KiY)
Head honcho, big cheese, top dog. An all-round nasty bugger. I swear that 2 Clue token removing Servant ALWAYS shows up when I least want it. Unless you're loaded with Items, be prepared to discard some decent ones. Oh yeah, say bye-bye to your Skills. Only managed my first seal on the 6th try (oddly enough, with some of the least favourite investigators: Vincent Lee, Amanda Sharpe and Gloria Goldberg; only redeeming one was Jacqueline Fine). Final Combat can be a surprisingly long affair, if the Terror Track is low to begin with. You can easily get 8+ turns to kills Glaaki, which is way more than you usually can rely on against other GOOs. Then again, failing Terrible Experiment and the 4 investigators having combined 3 dice is an experience I'll not forget (1 round of attacking before eveyone is devoured, 3 dice, need 48 successes, you do the math). Don't forget, Servants not yet in play are going to pop up during Final Combat so spread those items around.
Main reason I rate Glaaki higher than Yog is that Yog's impact in the game is a known factor, while the Servants are random and unknown until you place one. And I always draw Mythos card that would not open a gate/cause monster surge when drawing where to place the Servant. In one game, "No One Can Help You Now" lasted ages as I always drew an Environment card for the Servant and never for the regular Mythos. Both are really tough buggers though, I always know I'm in for a difficult game against those two.