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Experience Points and Uncontrolled Stat Growth
Strictly speaking these two issues, experience points and uncontrolled stat growth, are separate. But they are so closely interwoven in practice that I can only treat them together.

Uncontrolled stat growth means that there is no upper limit to how powerful characters can become. This is a problem because it allows characters to become more powerful than any character could reasonably get in that gameworld.

Experience points (EXPs) are the method used in most RPGs to increase the stats of player characters. The problems caused by EXPs vary from system to system, but most games have several problems:

(A) EXPs allow player characters to increase their stats at a different rate from NPCs: faster during gameplay, but usually not at all during downtime.
(B) In EXP-based systems, the stats that increase are often not those abilities which the character has been using.
(C) Player characters do things which the players know will get them EXPs for little risk or effort, though the characters would have no motivation for doing such things: they attack unsuspecting passers-by from behind, kill helpless animals in pens or cages, or set fire to buildings at night to kill the sleeping people inside.
(D) EXPs create the impression that the reason for playing the game is to increase your character's power, and that becoming more powerful is a proper reward for good play. I find this idea fundamentally objectionable.

The text below is from a review by Anthony Gallela of the D20 system Star Wars RPG in Valkyrie Issue 22 (Spring 2001).

There is an old saying in RPGing that goes something like, "Never give stats to God." This is assuming you would never want a character to have higher stats than God. Game designers continue to make this mistake, making punks out of our heroes. The James Bond RPG gave stats to its namesake, PCs quickly overshadowed him. Well, it's happened again in WotC's Star Wars. Maybe giving Watto or C-3PO stats is okay, but Yoda? Getting as powerful as Yoda affects a person's ability to suspend disbelief.

As soon as I read this I thought, 'No - this is wrong. The problem is not that very powerful NPCs are statted, but that the stats of player characters are allowed to increase unchecked beyond reasonable levels.' This is something which has long pissed me off and I often rant at great length about how much it annoys me.

Gallela is right that there is a problem with characters getting more powerful than Yoda, but his solution - don't stat Yoda - is ludicrous. The right solution is to stop PC stats from ever going that high.

Leaving Jedis aside, I'll go back to the old standard of swords-and-sorcery fantasy RPGing. Let's say for simplicity that we're playing a game like D&D or Dragon Warriors where characters have levels, and level corresponds pretty closely to fighting ability: so any two third level fighters will be roughly equal in fighting skill, and any sixth level fighter will well outstrip any third level.

This being a feudal hack fantasy world, the rank-and-file soldiers of the Watch are recruited with no particular fighting ability from local peasant stock, and then trained. A raw recruit has level 0, and might increase by around one level every two years. Thus a Watchman of five years' experience would typically be a second level fighter, and the grizzled old captain with twenty years' experience is a tenth level fighter. However, experience of war rather than the policing duties of the Watch would give one the same level of fighting experience in a shorter time, so a full-time soldier on campaign might gain two levels a year: thus a veteran of five years of war could be a tenth level fighter.

Nobles are trained in combat from an early age, so when a young nobleman comes of age he is already something like a fifth level fighter. His rate of improvement would be the same as for anyone else, dependent on the amount of combat experience he gets. If he went to war he would improve at the same rate as our peasant veteran above.

The hardest human fighters I'm ever likely to meet are the King's Guard, an elite legion of honour into which veterans of proven ability are recruited, and form the unit which the King himself leads into battle. These chaps have years of experience and are frequently in battle, and so the average member of the King's Guard is a twentieth level fighter.

Now we get to player characters. Suppose I'm playing a fighter who is a normal human, naturally quite gifted in the arts of bashing by virtue of being endowed with the proper physique and psychology for it, but without any kind of supernatural advantages - just a talented normal. I start the game as a first level fighter. I get in a fight with the tenth level war veteran described above. He knocks me around a bit but lets me live because he's sick of killing and just wants to live a quiet life now. I am a typical roleplayer and hold a grudge against his character which I nurse bitterly, plotting my revenge.

This being a D&D-style game, I quickly amass lots of experience points and go up the levels. In terms of game-time, my character increases by one level a week (I have seen this happen, because GMs tend to give out the same number of experience points per game session regardless of how much game-time it is supposed to cover). In less than six months of game time, my character reaches twelfth level. This done, I find my old enemy the tenth level veteran and brutally slay him.

How can this make any sense? This man increased his level as quickly as an NPC can, and it took him five years to reach tenth level. I surpassed that in a small fraction of the time. My character has improved in skill beyond all reason.

And my character can just keep on going. In under a game-year from starting out he can reach 25th level and beat up a couple of soldiers from the King's Guard without too much trouble. This is completely unreasonable and no GM should ever allow this sort of thing to happen. Unfortunately, they do all the time because the game rules allow it.

Games designers should see this problem and write game systems that don't work like this. Such games do exist: MegaTraveller, published as far back as 1987, has no experience points but allows skills to increase by experience gained (which takes years) or by formal training. This system is not perfect but it's a hell of a lot better than experience points.

Warhammer FRP also makes some effort in this direction: a character's stats can only increase above their starting value by a certain amount, determined by a character's career. Thus a character with a career allowing +20 Weapon Skill may increase his Weapon Skill by 20 points above the character's starting level and no more. The highest possible advance is I think +30, which puts a reasonable lid on development. Unfortunately, some of the stats have not been properly regulated and so a human (average Strength 3) can gain a Strength value of 7 (equivalent to a Greater Demon). So it's a good basic idea that has been poorly applied.

But most games that come out still have uncontrolled stat levels, and most gamers and designers don't seem to care how ridiculous they make the game.

See also Stat Change over Time.