Rome, as a city, is a pretty unique place without having to do something like chariot racing or gladiator fighting, which would be unique to yourself, but are tourist activities that many people have done (though I'm not one of them). Most of my suggestions are of the "things to see" variety, because that's why I go to Rome. But it might give you ideas for yourself.
I've been to Rome a handful of times, and there are still places that I haven't been to, that i still have on my list for the next trip. So I'm pretty sure you won't have seen ALL the stuff everyone lists. Looks like Nero's Golden House (Domus Aurea) is open again; if you haven't already planned to go there, it's worth your time (if you like ancient Roman history). We visited right after it first opened; it sounds like it's more developed (or cleaned out) than when we went.
Something I did in Paris that was very meaningful to me was to go to the location of a favorite painting of mine, Caillebotte's "Paris Rainy Day." Do you have a favorite artist that painted in Rome? You could try the same. Or if a favorite person or painter lived in Rome, you could visit locations or neighborhoods that were significant to that person.
Like movies? Like any movies set in Rome? There are all sorts of websites that list where various movies were filmed. I haven't done that in Rome, but our family has done that in New York, and my film-buff son really enjoyed it. This kind of quest would lead you to parts of the city that you might not get to otherwise.
I really like mosaics; on one trip, we tried to visit every church in Rome that had mosaics. Or you could try to view every obelisk in Rome (google it; people have made lists of their locations). Or visit 10 different gelaterias and decide which you like best.
Walk or bike or run down the Appian Way.
In terms of doing something other than looking, are you a runner? I am, and running up to the front of St. Peter's early in the morning, and around Piazza Navona, are favorite memories. I've alway through it would be fun to run around the Circus Maximus (which is now a park).
Enjoy!