This is one of the best answers there is.

Can’t believe no one has mentioned it yet!

Wikipedia puts it well,

# In mathematics:

One hundred eight (or nine dozen) is an abundant number and a semiperfect number. It is 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 Euclidean space, the interior angles of a regular pentagon measure 108 degrees each.

There are 108 free polyominoes of order 7.

In base 10, it is a Harshad number and a self number.

The equation results in the Golden ratio.

# Religion and the arts:

The individual numbers 1, 0, and 8 represent one thing, nothing, and everything(infinity).

108 represents the ultimate reality of the universe as being (seemingly paradoxically) simultaneously One, emptiness, and infinite.

In Hinduism:

Hindu deities have 108 names, while in Gaudiya Vaishnavism, there are 108 gopis of Vrindavan. Recital of these names, often accompanied by counting of 108-beaded Mala, is considered sacred and often done during religious ceremonies. The recital is called namajapa. Accordingly, a mala usually has beads for 108 repetitions of a mantra.

The distance of the Sun from the Earth divided by the diameter of the Sun and the distance of the Moon from the Earth divided by the diameter of the Moon is approximately equal to 108. It is claimed that the great sires of Vedanta knew this relationship and thus 108 is a very important number in Vedantic chantings.

Martial Arts:

Many East Asian martial arts trace their roots back to Buddhism, specifically, to the Buddhist Shaolin Temple. Because of their ties to Buddhism, 108 has become an important symbolic number in a number of martial arts styles.

• According to Marma Adi and Ayurveda, there are 108 pressure points in the body, where consciousness and flesh intersect to give life to the living being.
• The Chinese school of martial arts agrees with the South Indian school of martial arts on the principle of 108 pressure points.
• 108 number also figures prominently in the symbolism associated with karate, particularly the Gōjū-ryū discipline. The ultimate Gōjū-ryū kata, Suparinpei, literally translates to 108. Suparinpei is the Chinese pronunciation of the number 108, while gojūshi of Gojūshiho is the Japanese pronunciation of the number 54. The other Gōjū-ryū kata, Sanseru (meaning “36”) and Seipai (“18”) are factors of the number 108.
• The 108 moves of the Yang Taijiquan long form and 108 moves in the Wing Chun wooden dummy form, taught by Yip Man, are noted in this regard.
• Paek Pal Ki Hyung, the 7th form taught in the art of Kuk Sool Won,  translates literally to “108 technique” form. It is also frequently  referred to as the “eliminate 108 torments” form. Each motion  corresponds with one of the 108 Buddhist torments or defilements.

In other fields:

• An official Major League Baseball baseball has 108 stitches.
• In the Manga and Anime series Shikabane Hime Aka and Shikabane Hime Kuro (Corpse Princess), the shikabane himes must hunt 108 evil undead corpses in order to ascend to heaven.
• The atomic number of hassium.
• The number of Mbit/s of a non-standard extension of IEEE 802.11g wireless network using channel bonding.
• There are 108 Stars of Destiny to collect in the video game Suikoden distributed by Konami for PlayStation
• In the video game Legend of Dragoon for Playstation, there are 108 species; the 108th of which is the God of destruction. Its soul returns to the world every 108 years to find a host.
• 108 is the name of a community of and for open source developers, created by Red Hat.
• There are 108 cards in a deck of UNO cards.
• There are 108 Code Crowns in Digimon Xros Wars.
• Volume expansion of freezing water is roughly 108%.
• A vault in Fallout 3 full of Gary clones.
• In Batman: Arkham City, there are a total of 108 medals for each Batman and Catwoman to collect in the 12 campaigns of the Riddler’s Revenge. There are also several other references to LOST throughout the game.
• The Chinese classic 水滸傳 (Water Margin) featuring the 108 Stars of Destiny (As mentioned by Anne K. Halsall in the comments)

Source: http://en.wikipedia.org/wiki/108_%28number%29

Also see:
1. http://www.salagram.net/108meaning.html
2. http://www.3marg.info/blog/significance_of_no_108.shtml

# Now some c00L (made-up) illustrations:

1. Symmetry:

2. Concrete (‘con’tinuous+dis’crete’) design:

Sony VAIO logo is also an example of concrete design,

‘VA’ representing an analog wave and the ‘IO’ representing a digital binary code.
Link: http://en.wikipedia.org/wiki/Sony_Vaio#Etymology

3. Poker face: