75 ft for standard recreational skiing / wakeboarding. If you want to mono ski or wake board at a level then it's a different story.
Slalom Mono you need a proper rope with take offs to shorten as you improve and the rope itself must be of the right type to give the right amount of stretch as you power out of the turn buoy. At the shortest shortening the rope does not actually reach the turn at 90 degrees from the boat and you are reliant on your body length to get the ski round it. I never got to this stage as I am short
Wakeboard again, once you improve you will need a dedicated rope as they like minimal stretch when pulling for the wake.
Toys generally use 90 foot ish unless you have a wish to kill the rider dead then you should extend the rope as much as feasible to generate the whip
Barefoot is generally 100 foot lines again with minimal stretch and you will see that they use a wider handle for stability. i.e waist width as you hang on with both hands.
Trick skiing you need approx 60 foot but most importantly there will be a bridle forward of the handle to put your foot in for toe hold tricks. It's a good idea to have a release in the boat and an intelligent spotter for this style of trick unless you want your manhood stretched
Parascending is generally 200 foot ish of 14mm rope and it's a good idea to watch for the current as you prepare for take off unless you want the father / son / plastic dinghy combo to be launched in the air at the same time as we did in poole bay some years ago
Ski racing answered above and I have no idea as when I tried it I ended screaming like a girl and sold the ski the next day