A good friend of mine visited Japan recently and was kind enough to engage in some "slime watching" for me. She found UFO Catcher machines featuring slimes and even tried her hand at winning an elusive plushie. Her experiences are quite enlightening; with her permission, here are pictures and descriptions of what she witnessed.
The entire first floor of the Akihabara Sega Club was all UFO Catchers, with the largest selection of slime catchers of any arcade I've been in. (Then again, I've been in only 3 or 4 arcades there, so not as if it counts much, hehe.) There were large slimes, smaller slimes, tiny slimes, satin slimes, slime and DQ character keychains, and even those flat pillows... (I also saw "UFO Catchers" in other places -- not in Akihabara -- with those huge angel slimes and the slime backpacks... But they weren't UFO catchers per se; you had to play a roulette game that several times I was *thiiiiiis* close to winning a slime backpack...
UFO Catchers are HARD! Instead of having three prongs like the crane games we have in the U.S., these things have only TWO prongs. How's that?! And as I found out, there are lots of different types of UFO Catchers -- some are the simple two-button kind, where the first button controls the back-forward movement and the second button does the right-left movement. And of course you get only one crack at each button. But you gotta be careful and know exactly which button does what. Sometimes the first button will be back/front, sometimes it'll be left/right. That how I got messed up several times, because I didn't think to check. And the slime pillows one in Akihabara (yeah, of course I tried it! ^_~ ) had THREE buttons -- the first controlled the back/forward movement, the second controlled the left arm, and the third controlled the right arm. ARGH! I swear, if you saw me play that thing, you'd understand when I say those things are IMPOSSIBLE to win!

In the picture, notice that the girl managed to catch the babble. Not only that, SHE ACTUALLY WON IT!

(Besu's note: this image makes me want to dive into that UFO catcher and swim through slimes, like Scrooge McDuck swimming through his money bin...)
This was at a Sega Club that was about 5 mins away from our hotel. I took this on the same day we went to Akihabara. I actually got in trouble for taking this picture, hehe...
Additional advice: Games are usually either 100 or 200 yen for one try -- I haven't come upon a game yet that was more than 200 yen a pop. Buuuuut, there are many games that will give you a *free* try if you use a 500-yen coin. ...[F]or example, [one machine] was I think 200 yen, but if you use a 500-yen coin, you get 3 tries. For some of the 100-yen games, you can get 6 tries for a 500-yen coin. So my advice: have LOTS of 100- and 500-yen coins on hand because you never know when you might find an arcade. The arcade that had the DDR machine, we might have totally missed because it was below street level rather than at the same level as the rest of the shops.
My thanks to Daryl of Daryl's Library: Final Fantasy CDs for the descriptions and pictures. Be sure to visit her page for fantastic information on Final Fantasy music and importing.