Two of my kids have 3 grade skips between them, with the oldest in a dual enrollment program that let him start a college math major at age 14. The younger grade-skipped child is now in 7th grade.
My anecdotal experience is that the advisability of the skip depends on the child in many ways. Both of my kids were enthusiastic about grade skips, and this fact was a large part of success at the time. My eldest skipped 3rd and went to 4th with math acceleration, too.
A single grade skip didn't put my kids into a situation where the work was challenging, but again, the outcome here depends on the child. Extra math and LA acceleration helped them in that regard. I take the view that it cut the boredom from "awful and constant" to something that was less bad.
That said, we won't skip the younger one another grade because of difficulties related to age differences. This idea may not be popular here, but: multiple grade skips create real and problematic differences with grade-level peers who are much older. My eldest has a spring birthday, meaning that some of the kids in his class were 2.5 years or more older than him. It was hard for him socially in 8th grade, when he was still a little kid and the kids in his class were adolescents.
Cognitively, he was still ahead of them. But cognition is only a small part of social life, and while things got better in 10th grade, some of the kids didn't really know what to make of him, even in 11th grade. He's staying an extra year in the dual-enrollment program, which has been a big help.
My daughter is adamant about "no more skips," but doesn't regret the one she did at all. Things were a bit hard for her in 5th grade, when the girls in her class started maturing and she wasn't. She attends a school with a combined 6-8 class now (and 3 other grade-skipped kids), and feels comfortable. Even so, her difficulties have been much easier than what my son has experienced.
Again, the success depends on the kid. I personally wouldn't shy away from a single skip for my kids, who are outgoing, are very cool with change, and were in favor of the skips.
As for a skip when entering a school for gifted kids, again, it depends. My son's second skip was into a situation like that, and the work was both challenging and rewarding. That school closed and he was back to a regular school after two years, at which point he sleptwalked his way to the high honor roll. YMMV. It depends on the kid. But those two years in the gifted school were amazing. The dual enrollment program is also that good, but this is due in large part to the two people who run that program.
My advice: ask your daughter what she thinks about a grade skip. Have more than one conversation and give her time to let the idea sink in before forming an opinion. Check out the gifted school. Are there other grade-skipped kids there? How do they define "gifted?" With an IQ cutoff (120? 125? 130? Etc?)? With achievement and teacher recommendations? If the bar for gifted is set low, check the programs to see if it delivers on what you need for your daughter. Etc.
Try to see all sides of the situation and make an informed decision as best you can.
