Apart from the desi talas, there are other set of talas called "Marga Talas". These talas, in addition to the angams in the desi talas - laghu, drtham and anudrtham have other angams called Guru, Plutham, Kakapadam. The 108 talas and other groups of talas come under this group.
1 Guru - 1 beat and counting 7 fingers
1 Plutham - 1 beat, 1 krshyai & 1 sarpini
1 Kakapadam - 1 beat, 1 krshyai, 1 sarpini & 1 pathakam
1 krshyai - waving the hand towards left, it has 4 aksharams
1 sarpini - waving the hand towards right, it has 4 aksharams
1 pathakam - raising the hand vertically, has 4 aksharams
These talas are complicated and are found in very few compositions. In fact, the music of Tamils in ancient times had complicated rythm patterns like Chandha talam. Rythm was given importance. The Thiruppugazh is a classic example of the variety and complex nature of tala pattern in Carnatic music. The uniqueness of this tala lies in the fact that it varies according to the stress and rhyme-patterns (called Chanda) in the Tiruppugazh.
An example of a tala that uses all the units mentioned so far is Simhanandana tala, the longest tala, with 128 units.