Zip System Tape — The most expensive tape I own. It stretches like you won't believe, and the adhesive is almost as tenacious as contact cement. I carry some in my car for emergency repairs, especially as it's rated for outdoor use.
For vehicle emergency repairs, I also carry an exterior-rated duct tape, in case I want adhesive that will be easier to remove later.
medical tape — Designed to stick to skin, and to reduce the amount of pain upon removal. It's also designed to not leave much residue behind.
Some of the strongest you'd want to try is 3M Transpore tape, and any of the tapes they use to tape down IV lines (eg. 3M Tegaderm). These will definitely hurt some when removed, and will leave some residue, but they're appropriate in certain situations.
artist's tape AKA drafting tape — Good for temporarily taping down a piece of paper, and it won't damage the paper when removed. It's like masking tape with less-sticky adhesive.
Frog Tape — A type of painter's tape (the only painter's tape that's worth buying), used to paint clean lines. Its strength is the ability to be removed cleanly from almost any surface, which makes it useful in many other situations.