We initiate research on self-stabilization in highly dynamic identified message-passing systems where dynamics is modeled using time-varying graphs (TVGs). More precisely, we address the self-stabilizing leader election problem in three wide classes of TVGs: the class TCB(Δ) of TVGs with temporal diameter bounded by Δ, the class TCQ(Δ) of TVGs with temporal diameter quasi-bounded by Δ, and the class TCR of TVGs with recurrent connectivity only, where TCB(Δ)⊆TCQ(Δ)⊆TCR. We first study conditions under which our problem can be solved. We introduce the notion of size-ambiguity to show that the assumption on the knowledge of the number n of processes is central. Our results reveal that, despite the existence of unique process identifiers, any deterministic self-stabilizing leader election algorithm working on the class TCQ(Δ) or TCR cannot be size-ambiguous, justifying why our solutions for those classes assume the exact knowledge of n. We then present three self-stabilizing leader election algorithms for Classes TCB(Δ), TCQ(Δ), and TCR, respectively. Our algorithm for TCB(Δ) stabilizes in at most 3Δ rounds. However, we show that stabilization time cannot be bounded for the leader election problem in TCQ(Δ) and TCR. Nevertheless, we circumvent this issue by showing that our solutions are speculative in the sense that their stabilization time in TCB(Δ) (⊆TCQ(Δ)⊆TCR) is O(Δ) rounds.