DD, 13, skipped K-3 (well, okay, she effectively was homeschooled for preschool and kindy, and there we were at 6 yo)... compacted 4-5, and is currently "skipping" tenth grade.
She really hasn't "skipped" so much as been allowed to accelerate through the curriculum at a faster pace. She's actually completed GT/Honors curriculum via her public school for grades 3 through 10, mostly-- she's just done it in 6 years, not 8. So that amounts to a total of what, four skips? Hard to keep track of.
She'll graduate when she's fourteen. While we have some concerns about her maturity with respect to her ability to live on her own... she handles herself with her academic peers just fine aside from occasional wistfulness/angst that they are all driving and much more independent than she is (well, go figure-- they are mostly 16-18!); she fits in just fine and has made friends without much trouble.
One thing to be prepared for with multiple skips is the fact that emotional maturity and life-experience will of necessity lead to your child interacting with standard curriculum in some surprising/unexpected ways.
My DD was shocked by information re: eating disorders as an 8yo, and thoroughly grossed out by human reproduction in high school honors biology at 11-12 (she's decided that she is NEVER having sex or delivering a baby... LOL), and her second experience with Romeo and Juliet in freshman English (at 11) led her to the conclusion that Juliet is an manipulative brat who should have listened to her elders, and Romeo is a flake who is being driven by hormones. Completely irrational, both of them-- and look where it got them.
As long as everyone is prepared for that, multiple skips tend to go pretty well for children that are ready and able.