I have a long list since I was a small kid.
Reading was the first one, since I was hyperlexic, so all written material was just by its nature interesting for me. I would read all the books at home, plus all flyers, catalogues, free magazines thrown into out post box. When we visited my grandarents instead of talking to them I would read their stuff too.
I also remember being obsessed with the idea of queens and princesses. Most girls are, but I also had several notebooks that I filled with my drawings, always drawing the same female figure which was a queen/princess, just different hair style and dress. I would also make up different personalities for them.
Then it was astronomy for a while, and aliens.
I also liked quiz books and collections of quotations, I had several such books and spent a lot of time reading/memorising them. Also, Latin sayings. Encyclopedias, because I liked (still like) how they collect all the knowledge in one book.
After primary school I was obsessed with dogs for a few years (different types and breeds), and literature, especially certain authors. I would for example discover that I like an author, and read all the books written by her/him. Agatha Christie was a huge one for me, and I also have an obsession with detectives in general I think. (However, only discovered Conan Doyle as an adult and I never had a proper "Sherlock Holmes period", but I liked the series with Benedict Cumberbatch.) I would also read the same book over and over again if I liked it - I had a book about Ebola virus that I read at least ten times, but it happened with other books too.
I guess from high school I focused mainly on fiction - Harry Potter for example, and languages. Due to this I speak English (as a second language), French, a little Italian, and now learning Japanese, and of course obsessed with Japanese culture as well, particularly kanji.
As a young woman I had my "fashion period" - at one point I had more than a hundred and twenty pairs of shoes, fifty handbags and lots of accessories as well. I liked organising them into categories (day/evening wear, colour, season etc.)
When I was about 20, I discovered online fandoms and fanfiction, but even there people found it somewhat peculiar that I would read ALL the fanfictions about a certain character/pairing that I like. (Yes, it took an awfully lot of time, as usually there is hundreds or thousands of them.)
At one point, between the age 21-23 I had a really unhealthy obsession with food. I am not saying I was anorexic because I now see that it was just another obsession for me, but at the time I lost a lot of weight due to making - and keeping - strict dietary plans. I also liked counting calories and looking at ingredients.
Later on, starting from age 28 I discovered photography, but I don't think it is an obsession, feels more like a hobby and even made some money with it.
Writing - I have been keeping a journal since I was 12, but usually also have several notebooks for different purposes, have had a few blogs, still update one or two of them occasionally. I write stories too but don't always finish and post them, sometimes I am just happy playing them out in my mind. For example I like thinking about different versions/outcomes of the same situation (the "what if" factor). I am not sure if it is an obsession though.
Psychology and the brain in general is also an ongoing thing for me and I never really knew why, until very recently.
_________________
Being obsessed with Asperger's Syndrome is a very Escherian place to be at right now.