I feel the best system is GURPS, if you can get players who can get past the relative complexity. It has rules for virtually anything, which quite often prevents rule-fights since it's easy to stop at "it happened this way because this rule on this page says so." Depending how much of the system you elect to use, it also already has rules laid out for things like brilliant moments, doh moments, overexertion when you need to run just a little faster or hit just a little harder, what happens when you slept too little, what happens when you slept *way* too little... I could go on for a very long time, listing off the hundreds if not thousands of things that have not only been thought of but also thoroughly tested and balanced for everything from the daily mundane to the life-threatening.
I also like it because it uses a point-buy system rather than rolling for stats. I don't like rolling for stats because you'll always have at least one person who will keep rolling until they get what they want. Or if you tell them they can only try so many times, Bill Johnson becomes Bob Johnson, who is a strangely similar but *totally* different character that can make 5 new rolls.
The daring can take disadvantages, which get you more points to spend on the things you want but restrict or hinder you in some way. Maybe your stalwart warrior likes killing a little too much. Maybe your brilliant scientist thinks every alien wants a hug, no matter how many teeth they have. Maybe your priestess of the sun god is so devout because she's terrified of the dark. Or maybe it's something as little as a mage who doesn't like to get dirty but can deal with it... most of the time. It can range from a more-annoying-than-harmful quirk to a crippling flaw, depending how many points you want and how big or small a hindrance you're willing to put up with.
I like GURPS because I don't have to guess; someone else already thought of it and figured out exactly how to balance it within the system. Combat can be a little slow paced but I find it is far more satisfying because you have so many concrete options you can choose to take, both to attack and to defend. And there's a wide array of skills and tasks that aren't combat-related, allowing for a variety of campaign types.
It's also not tied to a genre. Low fantasy, high fantasy, steampunk, cyberpunk, modern, historic, science fiction, age of sail, and more. Take one, or combine elements of all. Fight aliens with your tesla rifle. Your bolt of lightning didn't work? Try your pistol instead. Or maybe you'd think your odds of survival better if you used a cybernetic heart instead of an organic one.
If you can think of it, GURPS probably has it all figured out for you. In short, I like GURPS because every player can be as imaginative as they want (within what you allow in your setting, that is) and everyone can be sure it's balanced and fair.