Part of the fun would undoubtedly come from the fact that you've created something that other people would find entertaining. Every person that plays video games, or just games in general, has probably, at one time or another, thought about working in the games industry. They want to make games and they want to make a game that other people would enjoy. The natural start of that career path would be to use an already-existing structure and create something awesome out of that, like the BGR team's Baldur's Gate 2 on the Dragon Age Engine mod. That could start with a mod, a map that can be loaded into the game by using something like the Hammer Editor for Half Life 2, or just getting down-and-dirty with some programming skills and making a game from the ground up, like these guys. The majority of people would avoid these routes completely -- they want something simpler. Most of us opt to create a game within a game (must... resist... Inception reference...).
The idea for your own little meta game might be something that you work out over many dark nights; some of the original 'zombies' modes emerged from, -gasp-, the imagination and lack of modding ability of FPS gamers. Other times, it might be something that you just happen to do once during the course of normal play and it happens to be fun. In fact, it's normally the spontaneous meta games turn out to be the fun ones, primarily because their uncalculated nature infuses them with a self-perpetuating randomness, resulting in more originality and uniqueness. Don't over-think it -- if you do, the once-spontaneous game will morph into a concept that's been done many times before.
One of the oldest ways to create your own game, or at least your own meta game, has been to create levels with an editor. For a while now this has been supported most often by first person shooters, but other genres have joined the fray on occasion. For years this was one of the best ways to get into the industry. Join a modding community, create levels, and get people to play them. The ideal objective was always to have a publisher take notice and getting your little modding team to turn the meta game you made into fully fledged title. Failing that, at least your level might have been noticed by the original developer and they may hire your services for future games. Either way, creating levels in an editor has always been a good first step and often shines brighter on applications than official qualifications.
One of the best places to go to create your own game has been Media Molecule's LittleBigPlanet, and especially its recent sequel. The first title in the series allowed players to create levels by using predetermined items, placing them in the level (with a theme or a story if you're so inclined) and then publishing that level to the community to be played, loved, and shared. The sequel allowed players to create whole games instead of just levels, with the addition of more electronics, better camera angles, and the ability to create cutscenes. What makes LittleBigPlanet such a good entry into a potential development career is the addition of the 'Mm Picks' feature. This feature allows Media Molecule themselves to look at levels and raise them higher in popularity than the other levels, which gets them played and shared even more. There's the added bonus that if your level becomes an 'Mm Pick,' you can be sure that the developers themselves have seen it. I know for a fact that people have gotten jobs at Media Molecule based on the quality of the levels they've created in LittleBigPlanet's create mode. So it's a route that's proven to garner results.
The easiest way of creating your own games is in the multiplayer mode of an already existing framework. Most games are entirely more enjoyable with friends, and creating a game that you can play with those friends only increases the enjoyment further. Maybe the game you create is to see how far you can launch a player using explosives, the player who get's them the furthest wins. Perhaps your idea of fun is to turn a game that usually doesn't rely on stealth mechanics into some sort of Metal Gear Solid clone. Either way, at that point you're creating a meta game, and it's the communities that develop around these ingenious uses of a game's multiplayer mode that thrive long after the game in question has stopped being played in the conventional manner.
At the end of the day there are many, many ways of creating your own games. Too many for me to really talk about all of them in length. Realistically, if you're interested in video games, then you're going to be better off choosing one of the methods I've talked about here. You'll be able to create your game in your own time without it impeding on anything else you're already doing, be that school or work. Once you're done there are already established communities where you'll be able to showcase your work and, especially in the case of LittleBigPlanet, easily get yourself noticed if you're good enough. If you want to work in the games industry as a designer, there's no better place than the meta game route. Creating your own levels and games. There are other routes for other careers prospects but I'll talk about those at a later date...