This is the classic [Conway's Soldiers](https://en.wikipedia.org/wiki/Conway%27s_Soldiers) problem, but not a duplicate of https://puzzling.stackexchange.com/q/12453 because diagonal jumps are not permitted. The highest reachable row above the line with a finite number of moves is the **fourth**.

The same weighting argument applies: a target square on the fifth row has weight 1 and all other squares have weight $\varphi^{\text{Manhattan distance to target}}$ (where $\varphi$ is the inverse of the golden ratio, not the golden ratio itself). No move can increase the total weight of all occupied squares, but the target configuration has weight 1, _as does the configuration where the whole lower half-plane is filled with pawns_. Thus no finite sequence of moves can bring a pawn to the fifth row – although [an infinite sequence of moves suffices](http://tartarus.org/gareth/maths/stuff/solarmy.pdf).