In this paper we address the question of the existence of a well defined, nontrivial fractal dimension D of self-affine clusters. In spite of the obvious relevance of such clusters to a wide range of phenomena, this problem is still open since the different published predictions for D have not been tested yet. An interesting aspect of the problem is that a nontrivial global dimension for clusters is in contrast with the trivial global dimension of self-affine functions. As a much studied example of self-affine structures, we investigate the infinite directed percolation cluster at the threshold. We measured D in d= 2 dimensions by the box counting method.