Traveling solo, wide open to whatever options. Physically challenging is fine as well.
My Iceland reccommendations would be as follows: Stop at the Blue Lagoon when traveling from and to the airport (most shuttles stop there on the way to the city). if you arrive on an early fight there is only a handful of people and it can be surreal.
The best option bar-none to see Iceland is to move out of Reykjavik and circle the island to see the many different type of landscapes. Rent a 4WD vehicle and stay in local farms that double as guest houses. (look for: "My travels to Iceland" by Dennys Bisogno on ibooks for one of the best travel guides I found for this purpose). If you don't have much time you can always use the guys at www.discover.is to drive you around in one of their tours (tip: further drives = better views)
For physical challenges:
-Dive at one of these two sites (the folks at ww.dive.is are great at organizing these dives): Silfra Crack - A waterway of glacier water that runs between the Eurasian and Amerian continental plates .
Strýtan - An underwater volcanic chimney..probably the closest one to the Surface (you may need a dry suit certification).
-You can hike at Landmannalaugar where the Laugavegurinn trail is (arguably) one of the best hiking trails in the planet...its 2-6 day hike and you will need to reserve huts and stop stations in advance. These trails close towards the end of September.
September might be early but you should try to catch the aurora borealis. You can even dogsled/snowmobile during the night under it. This website predicts when and where the Aurora will be most visible: http://en.vedur.is/weather/forecasts/aurora/
If you are staying in Reykjavik try the Kex Hostel to enjoy great afternoon drinks with plenty of other travelers. Laundry is a great water hole for local Icelanders. You must eat at Café Loki to try the fermented shark and wash it down with Brennivín..they even have a version distilled from wood!
Hope this helps
PS. I'ma long time fan of Tim F....Shout-out!
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There is one unique option I highly recommend. Do you remember Jules Vernes and his visions for the future? About a man going to the moon, Captain Nemo exploring the seas in a submarine. All of that was done last century. One of his visions was not accomplished though. The journey to the center of the Earth!
Now, for the first time, you can actually do that. There is a crater not far from Reykjavik called Þríhnúkagígur (Inside the Thrihnukagigur Volcano). It is a short hike to the crater and then you can descend in a basket 160 meters down into the crater. From there you can follow a tunnel into the magma chamber that opens up like a cathedral with amazing colours on the walls. This is the only place in the world where you can actually get into a magma chamber of a volcano through the crater. You'll find more information on www.insidethevolcano.com.