Which denomination(s) believe(s) there are many heavens?
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LDS theology explains the idea of heaven in terms of kingdoms of glory. The highest celestial kingdom being the place where God resides and the place where the faithful followers of Christ will dwell after the resurrection. LDS description of Heaven
Not a direct answer, but:
2 Corinthians 12:2 mentions "the third heaven". I think most Christians interpret this by a theory that says there are three heavens: the first heaven is the sky, i.e. where the clouds are and birds fly; the second heaven is outer space, where the stars and planets are; and the third heaven is where God lives. You can find plenty of references in the Old Testament where the word heaven is used to refer to each of the three, and it's the same word in Hebrew. I have never heard a Christian use this as part of a theory about multiple levels of the place where God lives, but maybe someone has somewhere along the line.
The Quran has a story about Mohammed being taken to heaven by an angel, and there are seven levels to heaven, with the most holy people going to the highest levels and the less holy people going to the lower levels.
There were some theories in ancient times about seven levels to heaven, based on the five planets known at the time -- Mercury, Venus, Mars, Jupiter, and Saturn -- plus the Sun and Moon.