One hundred and eight is an abundant number and a tetranacci number.

It is the hyperfactorial of 3 since it is of the form

108 is a number that is divisible by the value of its φ function, which is 36. 108 is also divisible by the total number of its divisors (12), hence it is a refactorable number.

In normal space, the interior angles of an equilateral pentagon measure 108 degrees each.

There are 108 free polyominoes of order 7.

108 is a Harshad number (Sanskrit: “Great Joy”). This means that it is evenly divisible by the sum of the digits, i.e: 1+0+8 =9. 108/9 = 12.

108 = 6² + 6² + 6², the sum of the squares of the three equal numerals making up the Biblical-Apocalyptic “number of the Beast”, 666. (just thought I’d throw that one in there… )

It so happens that the distance between the earth and the sun equals about 108 (actually 107-odd) times the sun’s diameter. Likewise, it so happens that the distance between the earth and the moon equals about 108 (actually 109-odd) times the moon’s diameter. That sun and moon look equally big in the earthly sky is the immediate result of their having the same ratio between distance and diameter. Moreover, it so happens that the sun’s diameter approximately equals 108 times the earth’s diameter.

Read more at: Why 108? by Dr Koenraad ELST, Ph.D., and at Wikipedia.