I can say without a lot of doubt that a lot of stupid things have probably resulted from the phrase "I bet you can't do that" or "I dare you". However, along with the stupidity often comes entertainment, and these terms are no exception in the realm of video games.
There are plenty of reasons why self imposed challenges in their various forms exist in the first place. Perhaps a person loves a game so much that they know its ins-and-outs by rote and despite still wanting to play it have become sort of bored by the normal way the gameplay unfolds. Or, of course there are the dares and bets of friends, or even that own voice in your head that goes "I wonder if it's possible to do this like this or if there's no chance at all". Sometimes people attempt the things they attempt to get recognition, or prove that their playstyle is viable or competitive or even superior to those that already exist. Sometimes people even just do it because they have nothing better to do and figure "hey, why the hell not?"
Regardless of the initial reasoning though, the player imposed challenge has given life to its own communities of players that love nothing more than trying (and often succeeding) at doing the seemingly impossible. These challenges have breathed life into older games, and helped allay some of the concerns that modern day gaming is "too easy" (a concern which I don't personally share, but that a lot of people seem to voice nonetheless). Whatever the premise behind the challenge though, the result is still the same: people hone and often times get to showcase their skills, and add hours, sometimes in the ranges of 10s or even over 100, to games that they've already played quite a bit of in the first place; in other words, people are getting more bang for their buck, and that's never a bad thing.
Some games of course already cater to a limited selection, thanks to things like difficulty levels that present more challenges and less room for error. To some though, even the most challenging difficulty becomes mundane after countless playthroughs and enough practice. That's where one of the most basic challenges comes in: playing a game (often on the hardest difficulty) with the starting weapon, or without upgrades, or even without trying to gain levels in the case of RPGs. Certainly, going through a game with the first weapon you're handed when you know there's going to be five to ten upgrades that are going to be offered to you is going to present a challenge: bosses that were already difficult even with the best gun or sword in the entire game turn into monumental tests of skill and stamina: knowing when to attack, when to back off, and how not to get yourself killed after whittling the boss into the red over the course of a half hour or more.
RPGs are even more difficult: how does one avoid random encounters and leveling up from the mandatory boss fights? There are guides out there for such low level runs that are massively complex and detailed. I personally remember the ones that sprang up for Final Fantasy IX after people learned that if you got to a certain point in the game - near the end of the final dungeon actually - within a certain amount of time - I believe within ten or twelve hours, which for a Final Fantasy game is kind of insane - then you could get an excellent weapon for one of the characters. People took the challenge not because the weapon was good (hell, arguably it is nowhere near worth it) but to prove that it could be done. So you had guides that detailed how to get through boss fights as fast as possible, how to make it from area A to area B in the least amount of time, even how to walk in order to avoid random encounters through complex exploitations of the games coding.
All of this amounted to one hell of a low level speed run community that centered around Final Fantasy IX. But of course speed runs themselves are nothing new, and are in fact one of the most commonly seen type of challenges: doubtless because under certain circumstances they can get you worldwide recognition up to and including a place in the Guiness Book of World Records. It's common to see "tool assisted" speed runs on Youtube, where through the use of various tricks that things like playing the game from a ROM allows, the player blasts through a game with lightning speed. While doubtless impressive, the unassisted speed runs sometimes even more so, and there have been some rather impressive ones: games being beaten in minutes that would normally take hours to clear, such as beating Quake on the hardest difficulty in less than 20 minutes. Records have been kept for both assisted and unassisted speed runs, and the amount of effort and planning that goes into either type is truly impressive.
Perhaps the ultimate extension of this are ROMs that are meant to test the absolute limits of player ability. ROMs like the infamous Kaizo Mario World series exist only to prove that with enough time and determination, a player can make their way through what seems like a ridiculously impossible situation. And despite the brainmelting, keyboard smashing frustration that must come from playing this kind of game, people do, and do so often. Because sometimes, really, all you need and all you want is a good challenge; sometimes that just means you have to end up making it yourself, one way or another.