It's very common to see a figure skater perform two consecutive jumps, and there are some who can add a third one. But I haven't seen anyone performing four consecutive jumps. Is it technically impossible? What's the problem with doing it?
The only limit is ability. I've seen skaters who can do dozens and dozens of double toe-loops in a row. I myself have easily done combinations of five or six jumps, and I'm not on championship level by far. But the harder to do the jump, the harder it is to chain it. The reason professional skaters don't do these combinations is because they won't get anything for them in competitions: Because of the rules they can only do two combinations of two jumps and one combination of three jumps in their programs, and there is a limited amount of total jumps in a program. If this limit didn't exist, skaters would just jump as many times as possible during the program and not do anything else to rack up their score, and thats not figure skating, thats endurance speed jumping.