I looked at my game collection for the weirdist games that I have and heres a small list that I came up with. Not including the other three I previously mentioned. (Basewars, Earthworm Jim, and Typing of the Dead)
Giants: Citizen Kabuto is one of the weirdest games I've ever played. I bought it from a 2nd hand store for a few dollars a long time ago. Having never heard of it, plus it only being a couple dollars really didn't set my expectations too high but the game is actually really good. To be honest I haven't played it in years so I can't remember a lot about it other than the fact that it is a 3rd person shooter, it has a good sense of humor and I think you can upgrade your bases and stuff, adding a RTS element into it. I had a blast playing the game. I mainly remember it being funny and challenging and incredibly different from anything else I've played.
Maybe this can tell you more about it but I highly recommend it.
http://en.wikipedia.org/wiki/Giants:_Citizen_KabutoDrawn to Life is a fun platformer where you get to draw a lot of the things that you interact with in game. You get to design and colour the main character, parts of the town, various platforms and vehicles. The game is has a nice charm to it and it is a lot of fun, but it is very easy and short.
Scribblenauts Most people know about this game already. A puzzle game where you get to choose which items you want to use to solve the puzzle. I found the concept to be very creative and amazing. I had a blast for the first few hours tying to come up with unique and quirky ways to solve the in game puzzles, but even though there is more than 22,800 unique words, the majority of them are useless. I found myself being too clever for the game, and a lot of my creative solutions didn't work simply because many of the objects and situations aren't programmed to do anything or interact as imaginably as you would have liked. I can't blame them too much for that though but I was still let down somewhat by simple concepts not working. Adding a helicopter or plane with a pilot next to it doesn't do anything, and other things such as shooting a mirror with a shrink ray to shrink yourself also doesn't work. Overall a good game but most of the puzzles will be solved by the same 30 objects or so. Rope, trampoline, rocket, explosive, any vehicle and all the variations of those can solve pretty much anything. The controls are also frustrating because you have to use the stylus to do everything, including moving your character and interacting with everything. This is very problematic but fortunately they are fixing that in the next Scribblenauts game.
Henry Hatsworth in the Puzzling Adventure A cross between a "Tetris Attack" style puzzle game and a platformer. Killing an enemy in the top screen (platformer) will send it to the bottom puzzle screen where it can be killed again, and for good in the puzzle mode. You can also activate items and fill up a "Tea Time" meter by clearing things in the puzzle. When the meter is full you get to turn into a Robot and kick a lot of ass. I haven't gotten too far into the game but so far it is really good.
Hatris A rip off of Tetris but you have to stack hats instead of blocks. This game is terrible and stupid.
Populous: The Beginning My favorite of the Populous series. I haven't played this game in probably ten years but from what I remember it is kind of an RTS style game, except the focus is on the main Shaman character. I had a blast running around using all the different elements and awesome spells to destroy your opponents. My personal favorite is summoning a locust swarm to wreak havoc on your enemies. Also sending a ton of guys into a little shack and watching it rock back and forth as it is destroyed from within is always a good time.
XIII A first person shooter with a comic book style to it. I don't remember a whole lot about it other than the fact that the game is very stylish and David Duchovny is the voice actor for the main character.
King of the Monsters A fighting game for the Neo Geo starring various monsters as the cast. I really liked this game because of how well it adapted giant fighting monsters into a good video game form. Sure there have been a lot of giant fighting monster games but King of the Monsters does it right. Most of the characters are a parody of a monster movie cliché. Playng as a Godzilla look-a-like and beating up a King Kong look-a-like is a dream come true for many fans of the giant monster genre. Watching the city get destroyed as you and your opponent throw each other around is just one of the elements that really helped polish this fighting game.
Xenophobe This game is pretty awful but I have spent countless hours at my cousins house playing it while I was younger, because it was one of the only games he had for the NES. Xenophobe is really bad and if you don't have a sentimental attachment to it I really wouldn't recommend it. The only redeeming factors for this game are the fact that it is two player co-op, and it brings back sweet nostalgic memories for me. Everything about this game is weird so it definitely belongs in this list.
Chu Chu Rocket For many this game is one of the first to come to mind when someone mentions the Dreamcast. The concept is pretty original, and the style is super weird, but I found this game doesn't really stand the test of time. In my opinion it is really easy and it is just too repetitive to play for an extended period of time. The multiplayer mode is kind fun, but its hard to incorporate a strategy or hold anyones attention for a long time, mainly for the fact that it is a huge shit fest for lack of a better term.
Mega Man Soccer I remember being incredibly excited for this game when I first heard about it. Being a huge Mega Man fan and also playing soccer for the majority of my childhood, this game was sure to be heaven in a small gray cartridge, right? Well not really. Mega Man Soccer isn't that great. Overall it is pretty hard and just not very much fun to play. Going throughout a tournament and adding the different Mega Man characters to your roster is pretty sweet though. Each of the characters in game has a special Super Shot to help keep the game interesting and bring more diversity between how the characters play. I still played the hell out this game as mediocre as it is. Mainly because my love for Mega Man was overpowering and the concept is still pretty sweet. I would straight up murder someone for a Mega Man Hockey though.
A Boy and his Blob A very unique game combining puzzle elements into a platforming adventure. You play as a boy who discovers a weird Blob. Feeding the blob jellybeans allows Blob to morph into different things to help you overcome different obstacles in the game. Depending on which flavour jellybean, Blob will morph into different helpful things such as a ladder, a rocket, an umbrella and even a hole for you to fall through. I loved this game when I was younger but I found it pretty hard, mainly because of how confusing the jellybean morphing is. I remember making a chart to help me remember which jellybeans do what, and I would really recommend doing this as it will greatly decrease your wasted jellybeans.
Thats all I got for now. This whole topic has inspired me to make a polished article outlining the best and weirdest of the weird. I will get around to that eventually. :p