I have a 6yr old daughter with Aspergers too. She goes to swimming lessons throughout the year, an athletics club during term time and an Italian opera workshop for 3 blocks of 8 weeks, over the year. Each of these classes is an hour each week, so depending on what time of year it is, she could have one hour or 3 hours per week of classes. The classes vary in cost, from around £3 for the athletics to £8 for the opera (USD 4.80-12.90). For me (and her) it's worth it. I get some 'me time' and she always wants to go to them. I've tried lots of other classes in the past and these are the ones that have stood the test of time. She doesn't always co-operate at the athletics, but we're persevering and she does love it.
As well as these paid activities, I also seek out loads of free activities, usually for the weekend. For example, on Saturday, our concert hall had a family fun day. There were loads of arts & crafts activities and opportunity to play musical instruments and join in with singing, etc. We also go to local museums, which often have great things on offer, for families with young kids. We also have some country parks nearby and we just go for a stroll/cycle. My daughter loves nature, so this is just perfect. When we go to the cinema, it's nearly always the weekend morning kids show. The film will have been in the cinema for a few months, but she doesn't mind at all. We get in for about £1.20 each. You needn't spend a fortune on activities. However, I might be a bit spoiled, as I live just outside Glasgow and it's culture enthusiast's dream. All of the council run museums here have free entry too.
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"We act as though comfort and luxury were the chief requirements of life, when all we need to make us really happy is something to be enthusiatic about." Charles Kingsley