What is the longest-lasting protein in a human body?
Crystallin proteins are found in the eye lens (where their main job is probably to define the refractive index of the medium); they are commonly considered to be non-regenerated. So, your crystallins are as old as you are! 

Because of this absence of regeneration, the accumulate damage over time, including proteolysis, cross-linkings etc., which is one of the main reasons why visual acuity decays after a certain age: that is where cataracts come from. The cloudy lens is the result of years of degradation events in a limited pool of non-renewed proteins.

Edit: A few references:

This article shows that one can use 14C radiodating to determine the date of synthesis of lens proteins, because of their exceptionally low turnover: Lynnerup, "Radiocarbon Dating of the Human Eye Lens Crystallines Reveal Proteins without Carbon Turnover throughout Life", PLoS One (2008) 3:e1529

This excellent review suggested by iayork (thanks!) lists long-lived proteins (including crystallins) and how they were identified as such:
Toyama &amp; Hetzer, "Protein homeostasis: live long, won’t prosper" Nat Rev Mol Cell Biol. (2013) 14:55–61