There are many songs that have given me goosebumps at some point or the other, but few that do that constantly.
Opeth - A Fair Judgement: Basically everything from Mike's short acoustic solo through the "leave it be" part all the way to the end of Peter's solo. Musical orgasm!
Opeth - Windowpane: The relaxation solo before they go into the opening riff in preparation of the final verse.
Opeth - To Rid the Disease: Totally evil piano kicks in. Enough said.
More Opeth moments.
Metallica - To Live is to Die: The melodic part right before the clean guitars kick in. Mike plays that clean riff during his riff game.
Metallica - The Outlaw Torn: People may say whatever they want about the Loads, but this song has amazing musical moments. Kirk's slide solo after James' volume-knob notes is stunning.
Swallow the Sun - The Empty Skies: 04:10-05:58. One of the most impressive moments of build-up and explosion I've ever heard in a song.
Hollenthon - Lords of Bedlam: I've heard the violins at the start of this song are taken from some movie soundtrack, but it doesn't bother me. Coupled with the guitars, it's amazing.
Draconian - The Gothic Embrace: They start playing "faster", the guitars cry and a choir chants in the background. Simply amazing moment right there.
Therion - Clavicula Nox: Lots of goosebumps over there. Especially around the 01:15 mark.
Blackfield - The Hole in Me: Chorus.