The Grundy number of a graph G, denoted by Γ(G), is the largest k such that G has a greedy k-colouring, that is a colouring with k colours obtained by applying the greedy algorithm according to some ordering of the vertices of G. Trivially Γ (G) ≤ ∆(G) + 1. In this paper, we show that deciding if Γ (G) ≤ ∆(G) is NP-complete. We then show that deciding if Γ (G) ≥ |V (G)| − k is fixed parameter tractable with respect to the parameter k.