Fast timescale state estimation for a large power system can be challenging if the sensors producing the measurements are few in number. This is particularly true for doing time-synchronized state estimation for a transmission system that has minimal phasor measurement unit (PMU) coverage. This paper proposes a Deep Neural network-based State Estimator (DeNSE) to overcome this extreme unobservability problem. For systems in which the existing PMU infrastructure is not able to bring the estimation errors within acceptable limits using the DeNSE, a data-driven incremental PMU placement methodology is also introduced. The practical utility of the proposed approach is demonstrated by considering topology changes, non-Gaussian measurement noise, bad data detection and correction, and large system application.