Wil,
'DON'T leave home without it!!!'
On our Alaska adventure a few years ago we were on the way from Tok to Chicken when some of the 'crush gravel' (it was the size of grapefruit and sharp!) spiked an inside on the duals. Slowly and with great care we made it back to TOK. There we found a fine gentleman to change the tire for us since I had a spare - otherwise it would have been at least a week to get a replacement tire.
When we arrived in Fairbanks, I purchased a replacement for the spare - at $25 less than the new original tires I bought in So Cal before we departed!!
Fortunately, it was not needed in the rest of the trip - but it was there! I think it is called 'insurance'.
So YES - carry a spare - mounted and inflated is best!!!