That depends on whether you mean how many different opening sequences or opening songs, and either way it's not simple.
There were three main opening songs. "Moon Over Parma," a song written by Robert McGuire and performed by Drew Carey himself, was the theme for the first season. It was replaced by The Vogues's "Five O'Clock World" for the second season, which was in turn replaced by "Cleveland Rocks," an Ian Hunter song performed by the pop group Presidents of the United States of America, for the third through seventh seasons.
During the final two seasons, the show switched between all three songs but not the original recordings. The openings featured different renditions of the songs by various artists, including a cover of "Five O'Clock World" by the pop punk group Bowling for Soup (who were a big deal at the time) and a sort of rap-lounge version of "Moon Over Parma."
And that's not even counting the unofficial fourth theme song, "Girlfriend in a Coma" by the Smiths. That one, a popular pop punk song from the '80s, was the intro to the two sixth-season episodes in which Drew was -- you guessed it -- in a coma.
If you're referring to the opening video sequences, rather than just the songs, it gets even more complicated. The show did have standard versions of each opening like a normal show would, with clips from past episodes playing under the credits. However, it also mixed that up frequently, either with different clips or even weirder choices (see, for example, the intro that played in season 8, which featured a bunch of dated-looking video effects to make it look like the old "Dallas" credit sequence).
All this chaos is what you get when you take an improv comedian and give him his own show, name it after him, let him hire his friends, and also make him head writer, producer and occasional director so that he has an incredible level of creative freedom. And make sure he has an offbeat sense of humor and a taste for obscure indie rock.
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