We have employed precise and carefully calibrated V - and I - band photometry of NGC 188 at WIYN Observatory to explore the cluster luminosity function (LF) and study the cluster white dwarfs. Our photometry is offset by V = 0.052 (fainter) from that of Sandage and Eggen & Sandage. All published photometry for the past three decades has been based on these two calibrations, which are in error by 0.05 ± 0.01. We employ the Pinsonneault et al. fiducial open cluster main sequence to derive a distance modulus of 11.43 ± 0.08 and E(B - V )=0.09, with the largest single source of error caused by uncertainty in the cluster metallicity. We report observations that are ≥ 50% complete along the main sequence to V = 24.6. We find that the NGC 188 central-field LF peaks at MI≈ 3 to 4. This is unlike the solar neighborhood LF and unlike the LFs of dynamically unevolved portions of open and globular clusters, all of which typically rise continuously until MI ≈ 9.5. Although we find that ≥ 50% of the unresolved objects in this cluster are multiple systems with mass ratios ≥ 0.3, their presence cannot account for the shape of the NGC 188 LF. For theoretical reasons having to do with the long-term survivability of NGC 188, we believe the cluster is highly dynamically evolved and that the low luminosity stars missing from the central cluster LF are either in the cluster outskirts or have left the cluster altogether. We identify nine candidate white dwarfs (WDs) in NGC 188, of which we expect at least three, and perhaps six, are bona fide cluster WDs. The luminosities of the faintest likely WD indicate an age of 1.14 ± 0.09 Gyr, where the error in age includes the cluster distance uncertainty and we assume the WD has a hydrogen atmosphere. This age is a lower limit to the cluster age, and observations probing to V = 27 or 28 will be necessary to find the faintest cluster WDs and independently determine the cluster age. While our lower age limit is not surprising for this ≈6 Gyr-old cluster, our result demonstrates the value of the WD age technique with its very low internal errors.