Today, I visited Amman itself starting by the Citadel. This was the center of Philadelphia, one of the cities of the Decapolis under Roman rule.
The Spanish-Jordanian team of archaeologs has been doing a great job there and they have been able to reconstitute a lot of the site.
There is an interesting and well restored Omayyad period palace and a roman temple (Dedicated to Hercules if I remember well).
The most interesting part of the visit is definitely the Jordan national museum that is located on the same site (entry fee included). It is a small museum but it is very well organized and packed with great objects. Following a chronological progression from the Paleolithic to the Islam period, the presentation is clear, well lighted and well documented. It is definitely one of my favorite museum so far.
Maybe because it is small enough to not exhaust you and diverse enough to keep you interested.
Then, I decided I had visited enough for the day and the heat (well over the 40C) was killing me so I found my way to the best big hotel swimming pool in Amman. Definitely worth it too ;).
A good easy day.