Polymers confined to small dimensions and that undergo high strains can show remarkable nonlinear mechanics, which must be understood to accurately predict the functioning of nanoscale polymer devices. In this paper we describe the determination of the mechanical properties of ultrathin polydimethylsiloxane (PDMS) films undergoing large strains, using atomic force microscope (AFM) indentation with a spherical tip. The PDMS was molded into extremely thin films of variable thickness and adhered to a hard substrate. We found that for films below 1 mu m in thickness the Young's modulus increased with decreasing sample thickness with a power law exponent of 1.35. Furthermore, as the soft PDMS film was indented, significant strain hardening was observed as the indentation depth approached 45% of the sample thickness. To properly quantify the nonlinear mechanical measurements, we utilized a pointwise Hertzian model which assumes only piecewise linearity on the part of the probed material. This analysis revealed three regions within the material. A linear region with a constant Young's modulus was seen for compression up to 45% strain. At strains higher than 45%, a marked increase in Young's modulus was measured. The onset of strain induced stiffening is well modeled by finite element modeling and occurs as stress contours expanding from the probe and the substrate overlap. A third region of mechanical variation occurred at small indentations of less than 10 nm. The pointwise Young's modulus at small indentations was several orders of magnitude higher than that in the linear elasticity region; we studied and ruled out causes responsible for this phenomenon. In total, these effects can cause thin elastomer films to become extremely stiff such that the measured Young's modulus is over a 100-fold higher than the bulk PDMS. Therefore, the mechanics of a polymer can be changed by adjusting the geometry of a material, in addition to changing the material itself. In addition to understanding the mechanics of thin polymer films, this work provides an excellent test of experimental techniques to measure the mechanics of other nonlinear and heterogeneous materials such as biological cells.