Let me use an example, and a recent one at that. I'm currently on my first run of Deus Ex. In the interest of testing the waters, I'm playing it straight-forwardly, all-violence and whatnot. And I'm on Easy. Don't worry, I have a stealthy and Medium save too, just this is for my first go. The most recent mission I did involved Juan Lebedev at the airport terminal. Now, in my infinite wisdom, I didn't realize that the Heal All option took most of your precious Medkits to heal your body parts. And seeing how I was on the last leg of the mission with my Head and Torso areas in the red, a single direct hit meant I was a goner. Deus Ex is already a game with multiple options to any situation, but as I locked myself into a set path, clearing out one guardhouse to get to the plane was a problem. I stood by the wall opposite the house's only entrance, and a few yards away in the corner of the area was a pair of cameras and a pair of turrets to go with them. Having already drawn the attention of some of the guards inside, I faced a difficult task: Kill the guards and not die. Yeah, simple isn't it? Even on Easy, the guards had no trouble killing me, and since my sniper rifle was out of ammo and I specialized in Rifles, the only good weapon I had was a shotgun and ... yeah ... so what I wound up doing that worked was, run blindly at the cameras and turrets. It was a guard tower with a locked door, so I hastily lockpicked it before the cameras turned the turrets on me, and I made my way upstairs and reprogrammed the turrets, taking out two of the guards. Turns out there were three people total, so the last guy put up little resistance. I'm amazed that the guards didn't just waste me while I was fiddling with the door.
The above is the difficulty I like. There is a challenge, but it's reasonably fair and the computer doesn't really cheat to piss you off.
A good example of the kind of difficulty I don't like is Prinny. Yes, I played it on Normal mode, with the three-hit clause, but I never finished it. Got frustrating. Yeah, I know it was marketed as a hard game and I knew that going into it, but something was just off. I didn't like how taking a hit caused you to bounce so much. How many times did I fall into a pit because of that? And having to completely restart on bosses was annoying too. Yes, I know older games did it too, but having to make the boss vulnerable first and then mash the  button during your limited window of opportunity? Ouch. Just let me flail away like normal.
Another good example of this is the final final boss of Dissidia Final Fantasy. A rare example of grinding in an RPG (or RPG hybrid as DFF is) not doing much. The boss starts at the unattainable level of 110, and this level gap gives him a nice advantage of starting with a higher Bravery amount, which equates to how much damage his HP attacks tick off. Well, the boss already has some pretty hard-hitting moves, though the first time you fight him, you probably had a great level advantage so you mopped the floor with him. This time, the tables have turned. The battle's the same, but the stats aren't. What was probably an easy go the first time around now is fairly difficult -- now you actually have to learn his attacks, and oh did I forget to mention that it's a three-stage battle that you have to completely restart if you die once? Your nerves get shot after a while, and seeing how the odds are stacked so heavily against you, it's not really reassurance that the game gives you. How disheartening is it to get to the last phase, even get the boss on the ropes...and a cheap shot takes you out. Granted, there are equipment options like two items that prevent death once (though one requires you not to be in Break status when the killing blow lands), but the bulk of the best equipment is gained after you kill him. This skirts the line between challenging and frustrating, though playing it constantly for two hours was probably stupid on my part. I did try creative ideas since I was playing under the idea that a single hit would off me, though of course I didn't even think to use the absurdly powerful once-use items. Whoops.
As I said before, I play games to have fun. There's a sense of accomplishment in beating a game, but there's accomplishment and absolute relief now that you've finally trudged through that one game. It's hard for me to describe, sadly. But I mean, I'm not the kind of person who plays games on their hardest difficulty for the hell of it. Everyone has their own brand of fun, and that's not mine. I remember playing the first Halo on Legendary. I got through part of the first level, up to when you had to climb up a staircase and there were plenty of enemies. I just never made it through there, even using all of the options I used, though that was years ago. I've at least heard of I Wanna Be The Guy's boggling difficulty, but that doesn't sound fun to me, sorry. There's only so many times I can throw myself at a wall, and I'll walk away before it cracks. Some people will keep going until they take down the wall. I just don't have that kind of patience.
Another thing that bothers me is trial-and-error gameplay. Related to the above, I can't try absolutely every possibility when it comes to options. Deus Ex gave me a pretty set few choices: Try to sneak behind people, try to take them on with a shotgun, or use external means to kill my enemies. Halo gave me few options: Try to take out the enemies on the stairs with a grenade, use the assault rifle...try not to die. I just hate when games ..well, not when the answer's not obvious, but when it's something that either will take eons to attempt, pure luck, or is something that'll come up later on to make the game impossible to finish. I somehow missed playing the early Sierra titles, where not taking some obscure item meant you had to restart the entire game. Extreme example, I know. Just, when you have to rely on supplementary materials to enjoy your game, that's not fun anymore. One good example I have is the final boss of Persona 3 (The Journey side, anyway). Apart from being a needlessly long and drawn-out battle, there is one phase where the boss can charm your team, which while bad enough in of itself, can totally screw your strategy. And this is even ignoring that the leader dying is an instant game over. Of course, there's a way to avoid the charm effect with an accessory, one of which you won't possibly know about unless you really experiment with your Personae, and you still have to luck out with that specific Persona being able to give that item in the first place...naturally, unless you read ahead or looked at a guide, you'd have to pray to the RNG gods that this specific attack wouldn't screw you over.
Speaking of the RNG...who here has played Fire Emblem? This is on the side of frustrating, but let's keep this going. Anyone who has played the game has likely balked at the battle status screen. "Oh, he has only a 3% chance to crit, I'm -- what why did he crit oh God he took out my Lord what the hell is going on" We've all been there. Of course, it could be bad luck, but oftentimes, the computer tips the scales into its own favor.. Okay, the rubberband AI thing is common in plenty of racing games, so I won't touch on that, but I do have a nice and recent example. I decided to start playing the PSP port of Final Fantasy Tactics. The first real battle you have, Gariland, is pretty standard fare. You have a team of weak newbies and so does the enemy team, but what really got me was how this battle went for me this time. I wasted at least five complete turns trying to kill the damned Chemist. Sure, he poses no real threat in battle, but every turn I had any damage on him, he'd use Potion and recover HP back to full. I had to double-team him to even stand a chance, and because the RNG hated me that day, I had to finish the battle quickly or be down two of my units in the first friggin' battle of the game! Infinite resources. Of course, this humorously comes back to bite the CPU in the ass later on when Auto-Potion will not heal more than 30HP per activation, and enemy Ninja units can potentially throw the best items in the game to anything with the Catch ability...
I'm actually kinda sad I missed out on God Hand. Trying a constantly-evolving difficulty system would've been interesting to see. The better you do, the harder the game is, so it's always "just right" in most cases. Or, that's how I've heard it. I do things on Normal/Medium/etc usually since that's meant to be the middle ground between Easy and Hard. If I plan on multiple playthroughs, then yes, I will go on other difficulties, but Normal is the "I'm going to play this once and this is what I should expect out of the game as a whole." And while I'm on the subject, what is with the idea that "harder" has to equal higher health/stats/etc.? Artificial difficulty in my eyes. Like, take a typical FPS game. Normal you can guess, but crank the difficulty to max. What changes? Usually, your attacks do less damage and your enemies' do much more to you. Oftentimes there's very little change of tactics, just a change in how dangerous situations get, and how long it takes to dispatch your foes. Diablo gives your enemies higher stats, more resists/immunities, and gives you a nice big penalty to your resists the harder the level (though the second game does balance this somewhat by having the best items in the game be available only on Hell mode). I do admit, just changing the stats does make the game harder, but what incentive is there to keep going? To say you beat it? That just doesn't work well with me, sadly. You can make the first Goomba in Super Mario Bros. invincible, but you'd just avoid it. Make all of the enemies invincible, and the game would be a little different. But what if that first Goomba grew big and chased you through the level? That'd be a massive change, sure, but if I play on Hard, I want a challenge that's outside of higher stats. Which takes me to another example. Yay!
Kirby's Dream Land. A nice game on its own, and beating the game once revealed the code to access Extra Mode, which while it was the same adventure, the enemies were different. They did more damage, but they behaved differently than their normal mode counterparts, and bosses were no exception. This is the kind of hard mode I like best. Yeah, it does have the higher-stat nonsense I mentioned earlier, but the enemies use different patterns and attacks. Things take you by surprise. It's actually a different experience, even if only slightly so. Kirby Super Star Ultra took this idea further with Revenge of the King mode, which was based on KDL's Extra Mode, and had the stages be completely different as well as use the new enemies. Not a lot of games I've played go to such lengths.
So yeah. A long-winded post that didn't really go anywhere, but whatever. I'm not a fan of games where pulling your hair out is normal. No, give me something I can enjoy playing. Even if it is easy. I'll take on a challenge, oh yes, but if it's going to be a waste of time pushing that rock up a hill only for it to keep coming back down, I'll just quit and leave it unfinished. Why, by now my backlog is so long that unfinished games don't necessarily bother me!
I will gladly take comments and questions.