THE SET of (based) homotopy classes of homotopy equivalences from a (pointed) space X to itself forms a group G(X), which is analogous to the group of automorphisms of a discrete group. It is of interest for many reasons and has attracted the attention of many authors. Recently, Sullivan[9] and Wilkerson[l l] have shown that if X is a simply-connected finite CW-complex, then G(X) is finitely-presented. Other authors had considered the structure of this group for various broad classes of spaces ([0], [3], [4], [lo]). In specific cases (see, for example [7], [8]), explicit computations show that G(X) is related to the group of units in a known algebra. In all these examples G(X) is finitely-presented. In this note, we give an example of a finite CW-complex X with infinitely-generated group of self-equivalences. We show that if X = S’ v S’ v S’ (more generally, S’ v S” v SzP-‘, p > I), then G(X) is infinitely-generated. Along the way, we need to analyze the group of self-equivalences of S’ v SZ (more generally, S’ v Sp, p > l), which is in fact finitely-presented. Thus our example is perhaps as simple as possible. Instead of considering spaces with few cells, another approach to finding a reasonable space X, with G(X) infinitely-generated, would be to consider spaces with a finite number of non-zero homotopy groups, each homotopy group finitely-generated and P,(X) finitely-presented. In fact, there is an example of a finitely-presented group 7~ for which Aut(n), the group of automorphisms of ?r, is not finitely-generated (see [S]). Hence if X = K(r, I), then G(X) is not finitely-generated. However, K(r, I) is a rather large complex, and there is no reason to believe that there is a small subcomplex Y of K(r, 1) with G(Y) infinitely-generated. Thus these two approaches are different. We work in the category of connected complexes with base-point, and base-point preserving maps and homotopies. When not necessary, the base-point shall be omitted from the notation.