Penetration by an object through a dense granular medium (for example, by a finger pushing slowly into the sand on a beach) presents an interesting physics problem that is closely related to issues of practical importance in soil science. Here we measure the penetration-resistance force for an object approaching the solid bottom boundary of a granular sample--analogous to the finger approaching a flat rock buried in the beach. We find that the penetration resistance near the boundary increases exponentially, which demonstrates the existence of an intrinsic length scale to the 'jamming' caused by a locally applied stress.