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We study online control of time-varying linear systems with unknown dynamics in the nonstochastic control model. At a high level, we demonstrate that this setting is \emph{qualitatively harder} than that of either unknown time-invariant or known time-varying dynamics, and complement our negative results with algorithmic upper bounds in regimes where sublinear regret is possible. More specifically, we study regret bounds with respect to common classes of policies: Disturbance Action (SLS), Disturbance Response (Youla), and linear feedback policies. While these three classes are essentially equivalent for LTI systems, we demonstrate that these equivalences break down for time-varying systems. We prove a lower bound that no algorithm can obtain sublinear regret with respect to the first two classes unless a certain measure of system variability also scales sublinearly in the horizon. Furthermore, we show that offline planning over the state linear feedback policies is NP-hard, suggesting hardness of the online learning problem. On the positive side, we give an efficient algorithm that attains a sublinear regret bound against the class of Disturbance Response policies up to the aforementioned system variability term. In fact, our algorithm enjoys sublinear \emph{adaptive} regret bounds, which is a strictly stronger metric than standard regret and is more appropriate for time-varying systems. We sketch extensions to Disturbance Action policies and partial observation, and propose an inefficient algorithm for regret against linear state feedback policies.