Why there are 108 beads in a spiritual rosary?

These are the steps to reach the supreme Lord Aksharateet Paarbrahm.

There are nine forms of bhakti or worship (navdha bhakti) in this world; shravana (hearing), kirtana (reciting), smarana (recollection), pada-sevana (service at the feet), archana (worship), Vandana (prayer), dasya (servitude), sakhya (friend) and atma-nivedana (surrender). 

The three paths of navdha bhakti are Jnana marga (path of knowledge), Karma marga (path of deeds), Rāja marga (path of contemplation and meditation)

So, 9×3 = 27

Now there are three attributes (Gunas)

That adds up to 27×3=81 (This takes one Up to the Vaikundd or /)

82^nd is Vallabhacharya’s method

83^rd is of Kabirji (This takes up to Yogmaya)

There are 25 aspects (paksha) of Paramdham.

The total now adds up to

83 + 25 = 108.