How many types of tape do airplane mechanics use?
I can't speak for General Aviation, but based on the number of tapes I've seen, it wouldn't be outside the scope of reality to think the aircraft manufacturers have a partnership with 3M to come up with as many different types of tape as possible.

For wiring repairs alone, I know our shop keeps at least 20 various types of tape on hand in our bench stock area to grab in a hurry. There are probably another 20-30 other types of tape we may use in rarer situations, but don't keep on hand. We'll usually order that on an as needed basis. 

There are a number of different types of "Speed Tape" as well. These are commonly used as very temporary structural fixes for small holes, dents, etc... on certain parts of the aircraft. It looks a lot like a sheet of aluminum foil, but it's some of the stickiest stuff I've ever dealt with. 

I know we also have different Mylar tapes that are used for a number of different applications. One of the applications, in particular, is when floor panels are sealed. The sealant can take 24-30 hours, in some cases, for a tack free cure. Since we want to use the airplane sooner, we can cover it with Mylar tape. It'll allow it to cure while under the tape. After the cure, we'll simply pull the tape up and continue business as usual. 

There are a number of other specialized tapes I've seen for various other applications. I do know that I have seen duct tape on bleed duct insulation. This is rare, but sometimes the insulation gets torn and people use duct tape to repair it. Quite frankly, I've never stopped to look and see what the approved tape is for that, but if there was one and only one application for duct tape, that might be it. 

I can't offer you an exact number, but they most certainly number in the hundreds if not thousands! Oh, and one other thing I've noticed is that Boeing tapes are usually different than the tapes specified by Airbus! So, each manufacturer has chosen certain tapes for certain applications and purposes, although some are interchangeable. 

If you really wanted to get crazy and technical, each tape comes in different widths too. All of these come in standard widths of 1/4", 1/2", 1", 2", and 3". I'm sure there are other width's too, but I know I've seen all of those sizes. 

Quite frankly, it's dizzying and unbelievable how much tape is on an airplane!