I had started wondering... what is your definition of "background character"? Lips is definately abackground character, as he has only had three known speaking parts. Thog is also mostly a background character, but he didn't start out that way. He was a main character in his first appearance, The Great Santa Clause Switch. He also had a lot of major roles before The Muppet Show, in The Muppet Valentine Show and several variety specials. He normally had speaking parts in those productions. He became a background character on The Muppet Show, where he only made occasional appearances, it was rare for him to make more than one appearance each season (after the first season), and he only spoke in two or three episodes of The Muppet Show. He has probably appeared in the same number of episodes as Lips, who debuted in the last season, and although both characters were in the opening, neither character was in very many episodes.
There were quite a few scenes on The Muppet Show that featured most of the full-body monsters, but Thog was usually left out, and he was usually the only full-body monster not to be included. The only times I know of that Thog appeared alongside other full-body monsters were in the episodes with Paul Williams, Bernedette Peters, and Alice Cooper. It is surprising that he wasn't made into more of a main character, like Sweetums was. I don't know if Henson was against having two major monsters on the show. However, I wonder if Jerry Nelson had trouble performing the character. Thog is one of the biggest monsters, and looks a bit heavy, though after I watched the first season DVD I started to notice that he doesn't really look that heavy to be inside. And The Muppet Show began around the time that Jerry Nelson gave up the role of Mr. Snuffleupagus because his back hurt from the role. Yes, Snuffleupagus is bigger than Thog, but I wonder if Nelson wanted to take it easy on performing full-body characters, and if Henson didn't want Thog to have a different voice (though a lot of characters were given new voices when The Muppet Show began. Nigel was now performed by John Lovelady instead of Henson, Janice was performed by Eren Ozker, and then Hunt, instead of Fran Brill, and Statler was now performed by Hunt as opposed to Nelson, so Thog could have been given a new performer, or he could have been like most of the other monsters and not had a consistent performer).
Also, the person who started this thread listed characters who aren't strictly background. Catgut, Baskerville, Nigel, and to a lesser extent Droop all started out as main characters, and Baskerville did have quite a few speaking roles, though most of them were in UK spots. And I don't really think of Boppity as a background character, either. He had a lot of speaking parts, and plenty of somehwat "starring" rles (interrogating Ruth Buzzi, talking about having three feet in At the Dance, making a few more significant At the Dance cameos, being one of the lead singers in Once a Year Day, being the first Muppet to join Gonzo and Candice Bergan in You've Got to Have Friends, being one of the three main monsters in There Are Millions of Us Who Are Ugly, and so on). Boppity has probably had more major roles than Behemoth, Timmy, Thog, Gorgon Heap, Gloat, and Beautiful Day Monster.