You are only allowed 2 stop-overs, and one destination ( which is your furthest point),
but you can have many connections under 24 hours. So if you find the right flights (
if they exist), you can get almost a day stopover in some cities with it actually
counting as a stop.

For example, I am flying a AE reward this week with a flight from CGK-TPE which arrives at 8:00 PM, then connecting to a flight TPE-ICN which leaves the next day
around 5:00 PM. So I am getting an almost full day in TPE without it counting as a
stopover. Bangkok is a good city to do this sort of thing in since there are a lot
of flights at all times of the day. HKG is another.
Going to Europe, for example, you could fly into FRA on the AC flight that arrives at 11:15 AM, and fly from there to BKK on the SK/TG connection via ARN that leaves FRA
at 10:40 AM - that gives you full day free stop-over.