Extend social skills, keep active and burn off some of the energy, keep fit, learn to play together and how to lose as well as win, learn some life skills, get different friendship groups, occupy her time after school (we both work FT and afterschool club is less boring if there is an hour of something between school and club), keep her out of our hair at times
DD has ADHD and Aspergers syndrome, which means she has bundles and bundles of energy and also has trouble making and keeping friends. But she is very sporty and also quite intelligent (factual not stealth boast - it's wearing in so many ways!).
So she has done swimming lesson when in crèche (Montessori) and we also did swimming lessons after school up to last year (important life skill).
She did Italian in crèche (Montessori) as 1 babyminder was Italian and gave Monti group an hour of fun/songs/games etc in Italian once a week. A minder in afterschool club when she was in P1/P2 was Spanish and gave optional Spanish classes once a week so DD did that as well. (Not that she remembers either, but when she hears it again she does understand).
Afterschool crèche also had computer class which DD did and loved. We didn't do others on offer there as she wasn't interested (other than playball).
She's always done sports ones. Playball was optional and became mandatory in afterschool crèche. Once she got to P1, school run loads of extra curriculars and she did athletics since then. She's done soccer for 2 years, and gaelic football. She just joined hockey this term (it was in a different school and we couldn't transport but juniors are now in their own school - we'll have a row next year about finding a lift).
School also have chess club and she's done that for a year (won both championships and beaten almost everyone so we'll give it a break for a while). She wanted to do lego club but that clashed with athletics and she wants to go to the schools championships there.
In our school, religious instruction is not done in school but catholic instruction is in school after hours so she does that (Church of Ireland is done in local CofI church and most other religions are done through home/place of worship).
She is also a member of the local gaelic club (gaelic football and hurling) so now does that 2 nights/week (was Sat am's and Wed evenings). Hockey club is Sunday morning (I was a hockey player until DD was 4 mths old). Sailing club is Sat afternoons (she loves water and boats) and sea cub scouts is on Saturday mornings. I grew up messing about in boats (rowing not sailing) and both DH and I were involved in scouting/guiding and enjoyed the adventures and learned a lot of useful things.
Apart from those, DD loves art and we do lots at home, but only do a few drop-in events in a local gallery rather than any formal classes. And baking/cooking. We all go hiking in the mountains, and DD does a fair amount of cycling (used to go to school with au pairs but not feasible at present due to different minders), and DH will take her to weekend clubs on bike a fair bit, occasionally I'll get on my bike too. She reads a lot at home but is not interested in joining school book club at the moment (clashes with hockey and chess). She has a guitar (I learned piano and guitar) and I am teaching her a little at home, but we can't find a teacher or time for lessons outside home, maybe she'll get into guitar class in school (very small numbers and very popular). We don't have room for a piano. She'd also like to do Coder DoJo but the local ones are full - there is talk of setting up one in school but needs more volunteers (DH has put his name down).
So there are lots of things we'd like to do beyond what we're doing - but at the same time, we are pretty fully extended. We have done some things to try, and dropped either because they weren't enjoyable or because we'd come on a fair amount and wanted other things more, or had to drop computers as we left the crèche (DD wanted to do lots of school ECAs so we put her into afterschool club in school - crèche only collected at end of school day not after ECAs so we were down to 2 days crèche and it was getting unrealistic).
We're also lucky that DD is our only DC, and we have enough to fund these activities. Our biggest enemy is time. But there are also limits to what DD can manage, and what we can manage outside of things available through school.