The Japanese Animation Club at The University of Illinois at Urbana-Champaign. JAC is one of the largest and oldest registered student organizations devoted to anime in the United States. The anime shown spans all genres (except hentai, which would violate student organization rules). Although an exact history of the club has not been maintained, flier archives for JAC exist as far back as September of 1993, when it was known as JACK: the Japanese Animated Cartoon Klub.
JAC holds its weekly showings in a large lecture hall on campus. All shows are open to the public, and no membership dues are required. JAC shows exclusively subbed anime, with a dubbed title being shown only once in the club's history as of 2001 (the Disney dub of Kiki's Delivery Service). In the past, JAC has shown series as diverse as The Vision of Escaflowne, Cowboy Bebop, and CardCaptor Sakura, long before the series came to the United States via syndication and DVDs. As of 2001, there were approximately 800 registered members, roughly 100 to 150 of which would regularly attend weekly showings.
But don't take my word for it; come down and see for yourself. Visit Loomis Laboratory, room 141, between 1:00 and 5:00pm on Saturdays during the school year. (On some rare occasions the meeting is in 151, or on Sunday. Don't give up right away!)
Edit: According to their website, the meetings are still at the same place, and the same time. How comforting that some things stay the same as time goes by.