What determines the shallowest depth in field in photography?

The largest aperture which is the smallest number on the aperture ring/dial generally anything between f1.2 - f5.6 dependant on the lens and zoom. On most consumer digital cameras the more you zoom in the higher the smallest aperture becomes.

eg. Lens at 35mm, widest aperture (shallowest depth of field) generally about f2.8

Zoom in and the largest aperture will reduce to about f5.6