First of all... awesome topic. I always talk about these 'moments of shiver' with guys at work... I didn't know if many people had these same feelings from TV, music, movies or books... it seems they do. In no particular order (and a mighty strange lot to be sure)... my shiver moments...
Transformers: The Movie
When Optimus confronts Megatron in the dark hallway within Autobot city... Megatron: Prime! Optimus: One shall stand, one shall fall. Megatron: Why throw away your life so wrecklessly? Optimus: That's a question you should ask yourself, Megatron. <--- insert ~shiver~ here.
Braveheart
When William Wallace comes back and gives Murron a thistle wrapped in cloth and she suddenly remembers giving it to him when they were children - and realizes he kept it all this time. Mass shivers also attributed to James Horner's haunting musical piece 'Gift Of A Thistle' that swells during Murron's realization.
Forrest Gump
When Forrest says the line, "Why do you hate me, Jenny?"
Willow
Whenever James Horner's main theme crescendos.
Rocky
When Rocky screams for Adrienne at the end.
Rocky IV
When Rocky is in his sportscar remembering Apollo through a series of flashbacks and the soundtrack song 'No Easy Way Out' by Robert Tepper is playing.
The Neverending Story
When Atreyu is screaming for his horse Artax to fight the swamps of sadness but to no avail...
The Lion King
When Rafiki holds up the just-born future king Simba and the soundtrack swells to the 'Circle of Life'