That being said, my first review is for a Sega Genesis game that's slightly more popular than most of the other games I'll be recommending. It's Toe Jam & Earl. Toe Jam & Earl is a very bizarre game to say the least. You play as two aliens (Toe Jam and Earl, shocking.) who crashed their rocket ship on earth, and they must journey across, well actually, I don't even know how many levels trying to find the 10 pieces of their rocket ship, which have pretty awesome names. I can't actually remember any of the names of the parts, but come on... They had a friggin boom box built into their ship. That brings me to my next point. The game has a terrific sense of humor. Toe Jam and Earl are the two funkiest aliens Sega ever created. Seriously, Toe Jam wears a huge gold medallion with his name on it.
There's also a slight RPG element to it. There are 9 levels your character can reach, and each one will give you more health, and sometimes an extra life. The titles range from things like Wiener and Dufus to Homey and Funklord.
Almost all of the other characters you encounter are "earthlings." Some of them help you, like the Wise Carrot (It's an old guy in a carrot suit), or Santa Claus. Most of them are out to kill you, though. They range from an angry lady pushing around a shopping cart and a guy pushing around a lawn mower, to some rather unearthly things like a giant hamster in a ball and a possessed, evil mailbox. (My favorite is the Phantom Ice Cream Truck.)
My very favorite thing about this game is the items. They all come in the form of presents, and since they're presents, you don't know what they are until you use them (After you've used it once, you'll know what it is... Until you open a Randomizer, which messes up your collection, and you have to identify them all over again.) Some presents are good, like Super High-Tops, which let you run really fast, or wings, which obviously lets you fly. Some presents are bad, like books. They make you fall asleep, and you need to start shouting "Wake up" until your character wakes up. Or sometimes your present will make it start raining tomatoes, or you'll open a Total Bummer, which kills you.
There have been two sequels to this, Panic on Funkotron and Toe Jam & Earl III: Mission to Earth, available on the Sega Genesis and Xbox, respectively. Panic on Funkotron isn't actually a sequel at heart though. It just borrowed the title, and replaced the platforming/treasure hunting on a 2D plane with an action 2D side-scroller. I haven't played the third version, but I believe it is more similar to the first.
This game shouldn't be that hard to find, actually, if you're interested in playing it. It's available on the Wii's virtual console, and most stores I've gone to that sell used Genesis games almost always have a copy of this game lying around somewhere.