We present an exquisite, 30-min cadence Kepler (K2) light curve of the Type Ia supernova (SN Ia) 2018oh (ASASSN-18bt), starting weeks before explosion, covering the moment of explosion and the subsequent rise, and continuing past peak brightness. These data are supplemented by multi-color Pan-STARRS1 and CTIO 4-m DECam observations obtained within hours of explosion. The K2 light curve has an unusual two-component shape, where the flux rises with a steep linear gradient for the first few days, followed by a quadratic rise as seen for typical SNe Ia. This "flux excess" relative to canonical SN Ia behavior is confirmed in our $i$-band light curve, and furthermore, SN 2018oh is especially blue during the early epochs. The flux excess peaks 2.14$\pm0.04$ days after explosion, has a FWHM of 3.12$\pm0.04$ days, a blackbody temperature of $T=17,500^{+11,500}_{-9,000}$ K, a peak luminosity of $4.3\pm0.2\times10^{37}\,{\rm erg\,s^{-1}}$, and a total integrated energy of $1.27\pm0.01\times10^{43}\,{\rm erg}$. We compare SN 2018oh to several models that may provide additional heating at early times, including collision with a companion and a shallow concentration of radioactive nickel. While all of these models generally reproduce the early K2 light curve shape, we slightly favor a companion interaction, at a distance of $\sim$$2\times10^{12}\,{\rm cm}$ based on our early color measurements, although the exact distance depends on the uncertain viewing angle. Additional confirmation of a companion interaction in future modeling and observations of SN 2018oh would provide strong support for a single-degenerate progenitor system.