How to write a for loop which runs an asynchronous command in each iteration?

Drop the ;:

for i in {1..8}; do sleep 100 & done

& separates commands, so the ; is extraneous (and the shell expects something between & and ;).

Group your do actions in curly brackets:

for i in {1..8}; do { sleep 100 & }; done

Easy to understand for you as well as for Bash!