Oh, there are so many ways games fail. Take MMOs. The way they're designed tends to make your character powerful in the very beginning, struggle for 10s of levels, and then in the end become a kind of godly character that destroys everything. And has to keep destroying everything in ridiculous number, because you have to kill literally thousands of enemies to level. The balance of the game continually changes which, though it keeps things from becoming stale, can have drastic effects on the way a class is played and in some cases really screw you over if, say, your spec is nerfed into oblivion and your gear set is made completely invalid. You can probably tell, I play World of Warcraft and am bitter about it. The way you fight in MMOs also bothers me. For the most part, melee combat tends to not be very interesting. It's just not interactive enough. You spam buttons, and there's really not that much thinking or skill involved. There's strategic use of abilities, sure, but there should be more than that.
I haven't really found an MMO I like pvp in yet. It's all bunches of people running around constantly, running through groups of people, frequently in some silly mini game like capture the flag. Where's the sense of realism? I admit to liking a fantasy background, but I prefer my fantasy worlds to involve things like realistic objectives, sound military tactics, collision detection, and archers not being able to hit you with instant-shot attacks with perfect accuracy while jumping and doing 360 degree spins. Abilities that make no sense for non-magical classes to have also bug me.
Large amounts of the game tend to be un-fun. When you're leveling you will run into all sorts of fun with questing, or running instances. In WoW, there's this dungeon where I kid you not, I have died more times after defeating the final boss due to enemies respawning on me on the way out than I have at every other point in the instance combined. Respawn rates are a horrible thing. If you need to kill a bunch of a certain type of enemy, it's usually too slow and you'll be hard-pressed to find enough of them. You have to go through a cave, kill some slightly more powerful boss enemy and get back out, the bosses respawn veeery slowly and you may have to wait several minutes for a chance to kill it... and then some other player may come up and attack it right when it respawns, and then you don't get the credit. Then, when you finally do kill it, you have to leave the giant cave full of enemies you don't have to kill who respawned while you were waiting for the boss. And you have to kill them, just to waste your time. Speaking of bosses, why is it that the only powerful bosses are in instances? Again, may just be a WoW thing, I've been playing it for entirely too long, but a "boss" enemy that you find out in the world at large is usually just one level higher than a normal version of that kind of enemy. When a quest tells me about this fearsome enemy that's butchered many people, and I go to kill it and find out that it's just very slightly stronger than it's minions, that's pretty anticlimactic. Enemies that are supposed to be powerful should be a challenge, not just your average encounter but with a name to go with it. And why, when I kill someone I don't have a quest for, can I not just go and turn in the quest for killing them that I'm going to get eventually instead of having to go kill them again in order to prove I killed them? It doesn't make sense, it breaks all sense of continuity. So does seeing them alive again 5 minutes later, for that matter. Of course, if you instance all bosses like some games do it removes that problem. Thanks some games!
MMOs also tend to be far too item-centric. I don't think your gear should be nearly as important as it usually is. It makes the game too focused on acquiring better items... so that you can acquire better items. When you're playing a game to get better items so that you can get better items, it loses a bit of its appeal. I would like to have games be slightly more creative than that.
I have also yet to see a magic system I really like for an MMO, but lets not go there.
RPGs: Well, RPGs are similar to MMOs in their way, except with more coherent storylines, goals, and endless puzzles
Puzzles make me sad. Especially jump puzzles. Why oh why do there have to be so many jump puzzles. Combat in RPGs tends to be better than in your average MMO, but they could still use some work. I wonder if anyone here has played Rune. Completely ignoring everything else for the moment, Rune had one of my favorite combat systems for an RPG so far. A reasonable selection of strikes, mostly making sense. Shields that you can actually block with, unlike most games where they just sit there and look pretty. It had some things I didn't like (instant kill attacks in pvp are kind of stupid), but all in all, it mostly worked. Still could have been a lot better though. Of course, I've not really seen magic pulled off right yet in an RPG either. D&D systems have a lot of good ideas, but they're too restrictive. We need a dominions RPG, you could get a pretty decent magic system out of that
I still won't be completely happy though until I'm playing in virtual reality, with complete control over every action my character makes.
Strategy games: There are several things modern strategy games do that annoy me. One of the main ones being too much focus on special abilities! In some games, practically every unit has a special ability of some sort. And 90% of the time you're better off manually triggering that ability than letting the computer do it for you. Now, sometimes you can pause a game to do multiple things at once, but this doesn't work in multiplayer and ultimately isn't the best solution. Furthermore, it's too much work. If you want to have a few things with special abilities, well, okay. I can live with that. But you shouldn't spend all your time dealing with special abilities.
I also don't like having to manage resources, production, and combat all at the same time. I can't watch everything at once, and I like to watch the combats and try to organize them. But if you stop paying attention to one thing in favor of another, you're going to suffer for it. You'll run out of resources, or your opponent will pull some trick against your army, or you'll stop producing units... and you can't watch everything all the time. It's just frustrating. And then combine that with special abilities on all your units, oy.
And then there's the fighting itself. There's too much emphasis on pulling silly tricks that take advantage of the mechanics of the game, and using special units with powerful abilities to ruin your opponent without letting him fight back. I want there to be more relevance in maneuvering, terrain, tactics! Like the Total War series (though I think I actually prefer medieval total war to medieval 2. Haven't played the rome or shogun series). If only they didn't rely so much on powerful generals artificially increasing the strength of your armies, it would be perfect.
Anyway this is getting pretty long so I think I'll stop now
Edit: AI design tends to hamper strategy games a lot more than most other game types. Probably because they have to do so much more. In some games it more or less works, but in some games it completely fails. Like in supreme commander, where basically your enemy AIs do nothing but send experimental units at you and build tons of defenses.