Why do we chant mantra for 108 times?

Mantra is a sound energy that cleanses the mind of desire. It churns the Mind.

One of reason is that there are 27 nakshatr / star constellation and 4 pada /charan (division of nakshatra into 4 sections are called padas / charan). The 27*4 = 108. So, we repeat the mantras for 108 times.
Nakshatras
Ashvinī
Bharanī
Kṛttikā
Rohinī
Mrigashīra
Ārdrā
Punarvasu
Pushya
Ashlesha (nakshatra)
Maghā
Pūrva Phalgunī (Pubbha)
Uttara Phalgunī (Uttara)
Hasta
Chitrā
Svātī
Vishākhā
Anurādhā
Jyeshtha
Mūla
Pūrva Ashādhā
Uttara Ashādhā
Shravana
Dhanistha
Shatabhisha
Pūrva Bhādrapadā
Uttara Bhādrapadā
Revatī

Another reason is that diameter of sun * 108 times gives the distance from earth to the sun.
Also, diameter of moon * 108 times gives the distance from earth to the moon!!