They exist. You may not know where to look for them.
Odd-time signatures (rhythms in 5, 7, 11, 13, etc.) are common in certain cultures. Odd-time signature songs and dances can be readily found in folk music from places like Greece, Bulgaria, and Turkey, and other nations in that geographic region, going back to antiquity.
The use of odd-time meters and frequently-changing meters in classical music was firmly established in the early 1900s, with composers like Igor Stravinsky and Béla Bartók.
In the USA and England and in the West, there has been a lot of music written in odd time signatures since the 1960s, in the styles of what we refer to as progressive rock, or progressive metal, which make use of odd time signatures for sections of the song.
A famous example is the 1968 piece by the Allman Brothers called "Whipping Post", whose main riff is in 11/8 time.
In progressive rock songs, there may be several different time signatures used in the same song, changing frequently. The band Rush is known for this, as are Genesis, Yes and King Crimson.
The main sections of the international hit song "Turn It On Again" by Genesis is in 13/8.
In the Yes song "Siberian Khatru", the introduction is in alternating measures of 8/8 and 7/8, creating the feel of a 15/8 time signature.
Peter Gabriel had a big hit in the 1970s with the song "Solsbury Hill" which is almost entirely in 7/4 time.