What is the complexity of depth first traversal that don't label nodes as discovered?
For general DAGs, the runtime could indeed be exponential in the depth. Here's an example of a family of such graphs. Let $G_d$ be a digraph comprised of a vertex stacked on a collection of 4-cycles as below, where I've illustrated $G_3$:



In this figure, the edges are directed from top to bottom and every vertex has two children. The number of function calls your algorithm makes (and hence the run time) for a digraph $G_d$ of depth $d$ will satisfy $T(0)=1, T(d)=2T(d-1)+1$, since every $G_d$ is comprised of a root vertex at the top with two overlapping copies of $G_{d-1}$ below it. This recurrence has solution $T(d) = 2^{d+1}-1$, exponential in the depth.