Looking at the comments today, I have to throw out Guns and Roses. They were only really at the top of their game from 1987 to 1991; Nirvana's Nevermind came out exactly a week after the Use Your Illusion double album. I do have to look at Van Halen. My problem with them is that their sound was so drastically different with Hagar that they almost don't feel like the same band. We're going to throw the Gary Cherone "era" in with Rocky V and Superman IV in the dust-covered trunk of things that never happened.
The Beach Boys were great, but they don't hold up as an epic band, with all due apologies to John Stamos. Dude, the fat kid from Stand By Me is sleeping with your ex-wife (I stole that joke).
We all sleep on Chicago a bit and the Jimi Hendrix Experience gets discounted because they're hardly remembered as a band, but it does bring me to one that we completely overlooked: The E Street Band. Do they count as a band? I think Springsteen is generally considered a solo act, with a back-up band, but that band has been together for a really long time.
All in all, I think I have to agree and stick with Aerosmith. Long career, ups and downs, the epitome of rock stars. Even with the crappy Armageddon song, they have to be it. Where would they rank on the British list? With Led Zeppelin? Right below them?
On Marissa's comment, this wasn't about influence so much, but Buddy Holly should definitely get the honorable mention. If we're talking influence though, I'll go with a most underrated influential band -- the New York Dolls. They were punk before punk, glam before glam. They bred the Sex Pistols, the Ramones, REM, the Smiths, Bon Jovi, Guns and Roses, and so on. They need to be in the Hall of Fame for their influence alone. From 1973, the New York Dolls (and yes, their lead singer was David Johansen of Buster Poindexter and Scrooged fame):