Good question.
Mostly I think it depends on quite how much in the way of rules and/or ease of play you want.
>> For something quick, expandable and effective there's the
Savage Worlds Revised edition - hardcover from Great White Games (a simplified & expanded version of the Deadlands system). All inclusive, plus plenty of free stuff (monsters, etc) can be found online. Uses different size dice (but only one at a time) to represent greater skill, all having to hit 4+ for a success, with greater degrees for every 4 pts over. Pretty easy to come up with new talents and add in extras for make things interesting. Has enough info in the book to represent most things initially. Fast and furious as the blurb says. Lots of free conversions online for many genres/systems.
>> For a system that's a bit more inclusive, with some multi-genre crunch, there's the
Fuzion system from R.Talsorian. Mixed quite heavily as it is now with the old Dream Park system it caters for a lot of options, though requires a touch more reading and digestion. Points based, including buying gear which could be an issue. Talents & Perks for more customising. Tons of equipment and weapons across a fair few genres/time periods. Also has a appropriate selection of clerical and regular magic, psionics and superpowers. Can buy pets, animal companions and vehicles with points too. Fairly expandable. Plenty of available free conversions online.
>> For the full on DIY approach there's the
TriStat dX system from Guardians of Order (free as a pdf from drivethrurpg as well I think). It's points based (similar bonuses and probs as above), though you have to have a fairly solid concept in mind when gen'ing as there's so many options listed otherwise things tend to end up pretty high powered no matter what the genre. Stems from anime and mostly better for higher end and simpler stuff. Doesnt have much in the way of campaign examples, no monsters, etc so will need more work out of the box. You can use several different ways to resolve conflict, skills, etc with one die (the default is normally d10 + skill).
>> Slightly less genre-vague is the
d6 System. There's the new versions out recently including d6 Space (any sci-fi plus psi), d6 Adventure (anything modern-esque, incl. psi & some magic) and d6 Fantasy. You may also be able to get hold of the original d6 Customizable System book, but that requires a lot more work across the board. Overall the system is pretty simple, has a decent amount supplied for each genre type and the info in each book is mostly interchangeable which means decent available resource. You can easily build spells, powers, monster abilities, etc from scrach using the same scale of x number of d6, roll and add up to achieve/exceed a target number. Lots of free stuff online including conversions for almost any genre/film/system.
The added bonus with the TriStat and Fuzion is that you can get free versions online and pretty cheap for a hardcopy, so can't hurt to snaffle 'em for a look if you're so inclined.
On a last note I'd also include
Everstone in there as a customisable base to work from. It's not actually a generic system (having its own unique setting), being a combination of stripped down d20 (new race/spell ideas) and BESMd20/TriStat above. Its points based, uses a nice race specific hit dice mechanic, an energy pool for spells and most abilities, is points based, and personally I think is very good for adding in new abilities, spells and stuff for classes, races and whatever. The possible down side is that it's still loosely d20, loosely level based (though this can be ignored completely if desired) and has its own setting (though also pretty good, specially for examples of powers).
Most of the systems mentioned can result in fairly good interesting, useful, self-contained characters from day one. You're unlikely to be worrying as much about keeping PC's alive early on as you would starting at low level with D&D say. Tends to make the games more fun and character driven in my experience. The systems tend to have options - and I love options (dont we all?).
If all else fails free to drop me a mail as I've got a couple of generic systems I use that may be of use
Cheers
Klayv