We've managed some radical acceleration by homeschooling with a local learning center. (They are excellent BTW, so if you are in the PDX area, PM me for the lowdown.) DD is currently around 3 years accelerated and we've seen both negatives and positives.
DD would be entering 6th grade in public school, but is about to start with a mostly high school level curriculum (with a few upper middle school classes for various reasons). The school situation (which I appreciate is unique) has provided mitigation for some possible negatives. This past year it was middle school level classes.
On the positive side, she's much happier being able to learn new things and actually having to work. Regular school wasn't good for her intellectual or emotional health. Her work load last year could have been more rigorous, but this year should actually be a challenge (AP Biology!).
Because the classes have broad age ranges, she get's to be the youngest on a continuum, rather than the youngest by several years. This helps keep her from feeling like too much of an outlier. Or rather it did. This year she she will go from being the youngest in the age band to younger than the age band and probably will be an outlier in some classes (though her biology last year had a 9-year-old so...).
Because the school has mixed ages, grades, everything, she can easily connect to a peer group with more similar ages (for her this means kids 1-2 years older, which is where she's comfortable). It also helps with the flirting/dating issue (which she is only just dipping her toes into). We've made it clear that any boy more than a year older is TOO OLD at her age. She's comfortable with that too. Basically, she's not cordoned off from kids her age.
Another good thing about the school is that they are super flexible about homework and grades (optional and nonexistent). This might be a negative for some kids, but for DD it's perfect. She can learn the material without being crushed by a too-heavy workload or too much need for executive function skills (which are not at all advanced in my DD). We do insist on her doing homework and give her grades (at her request actually, though you should have seen her face when I gave her a C!). The negative is that college might crush her... but more on that below.
Honestly though, I don't know that a straight acceleration in public school would have worked as well. Academically it'd be fine, but the social issues and PE and workload might have been too much for her.
Sports are all outside of school and are run by age. Many kids here play outside of grade if they are very good and there are many other's who've been skipped a grade. So the disconnect isn't too bad. DD has found ways of dodging school talk, though sometimes it bums her out.
Summer camp is by entering fall grade, but we naturally enroll her at the appropriate age and not grade, which is what they really mean anyway. She has permission to tell any story she wants to the kids at camp about school. She can describe herself as a homeschooler without a grade or as attending a private school. Since the assumption is that all the kids are entering the same grade, it usually doesn't come up.
There's a particular early admission program she has her eye on, but otherwise we are taking it year by year. It's given her some focus to have that as a goal and the process of applying (taking the SAT and ACT, essays, campus visit) will give us all a sense of whether she's ready or not. That will start next year. She will also take her first community college class in spring of next year and this should help her dip her toes into college level workload. We are looking at a science class with a lab. I realize that CC is not the same as a 4-year in terms of rigor, but it's a start.
As she ages there seem to be more and more options. For example, she could spend a year abroad. She can decelerate to attend a rigorous boarding prep school. She can go wider for a while and explore more interests through high school level classes and community college, without necessarily launching on a full-time degree track. She can do an online / middle college option (classes at the CC, but enrolled as an official high school student). She can attend college part or full time and live at home. If there's a need to slow her down, there are more ways to do so without it feeling like a retread.
I wish I'd had the opportunity to accelerate (or really to skip high school entirely). School was mostly a boring kind of torture and there was enough of a disconnect between me and most other kids that it wouldn't have been worse if I'd have been younger. Mostly I skipped class, doing just enough for the grades. My sophomore year in college was a bit of a shock, being as it was the first time I needed to expel effort.
