Yes, of course! As "fancy" coffee company Counter Culture says: Any Coffee, Any Brew. In fact, even things that aren't actually coffee (barley, tea, herbs, etc.) could be brewed like coffee in an AeroPress or other coffee-brewing equipment.
But if fancy coffee is not your preference (or not practical, or not available, or whatever), try something different. And why not a cultural staple like Maxwell House (or Folgers or Nescafé or Eight O'Clock or Café du Monde or ...)?
In fact, a mass-market coffee like this might benefit from AeroPress treatment. For example, some of these blends include a significant portion of Robusta beans, which is a traditional component of many blends (including some Italian espresso blends). Robusta might have some advantages in an AeroPress. Robusta produces more crema in espresso, and might be a way to get more crema from AeroPress. Robusta also has slightly more caffeine than Arabica, so that's something else you might be looking for. Since AeroPress (generally) infuses for less time than drip, you might reduce a bit of the bitterness sometimes present in mass-market coffees (regardless of bean makeup).
There might be a few problems, though. First, the default grind of pre-ground coffee like Maxwell House is rather course for an AeroPress. You could try to re-grind the coffee to be a bit finer, but it's generally not recommended because it can clog grinders (see more on this from Home-Barista and Bunn). Second, if there are other "stuff" in the coffee (e.g., "Folgers Crystals" or other additives) this might not behave the same way in the AeroPress as it might for drip.
But give it a try! In the best case, you can enjoy a mass-market coffee even more than you'd expect. In the worst case, you're out a few quid, but you have a fun story and a hefty caffeine buzz.