A downhole shock absorber preferably is filled with well fluid in a chamber that is reduced in volume due to impact. A rupture disc can hold the initial non-compressible liquid charge until impact. Upon impact, the rupture disc breaks to allow the fluid to be forced through an orifice to absorb some of the shock that occurs when a string hits a fixed object after dropping in the wellbore. The nose of the shock absorber is a soft material that has voids so that the combination of the softness of the material and the voids allow the nose to reshape until it encounters a surrounding tubular wall and then to collapse inwardly into the voids, making it simple to remove. The nose is releasably mounted to the shock absorber so the two can separate if the nose gets stuck after impact.