Let A be a finite dimensional associative algebra and d an integer, let gA(d) be the number of inequivalent indecomposable representations of A of degree d. We shall say that A is of strongly unbounded (representation) type if gA(d) is infinite for an infinite number of integers d. In this paper, we are concerned with the structure of algebras of strongly unbounded type. Theorem 2.1 states that an algebra is of strongly unbounded type if it has an infinite ideal lattice. This is a generalization of a theorem of R. M. Thrall which generalized the condition given by Nakayaina in [4] for an algebra to be of strongly unbounded type. In Section 3, we associate a graph with each ideal in the radical. Theorem 3.2 states that if any of these graphs has a cycle, a vertex of order four, or a chain that branches at each end then the algebra is of strongly unbounded type. The cycle and vertex of order four conditions are generalizations of conditions given by R. Brauer [2]. The branching chain condition is a generalization of a condition used by Thrall [6]. Two other conditions were recently obtained by T. Yoshii [9]. The function gA(d) can be used to define other classes of algebras. We say A is of finite (representation) type if Ed gA(d) is finite. Also, A is of bounded (representation) type if there exists d0 such that gA(d) = 0 for d ? d0 ; A is of unbounded type if not of bounded type. Nakayama first noted the existence of algebras of unbounded type in [4]. Concerning these classes of algebras, R. Brauer and R. M. Thrall have conjectured that algebras of bounded type are actually of finite type and that (over infinite fields) algebras of unbounded type are actually of strongly unbounded type. D. G. Higman [3] has shown that a group algebra is of unbounded type if and only if it has a non cyclic Sylow pgroup, p the characteristic of the field. T. Yoshii has confirmed the first conjecture in the case that the field is algebraically closed and the radical squared is zero [10]. Using the same assumptions, in [8] he obtains necessary and sufficient conditions that an algebra be of bounded type. Section 1 contains preliminary results. Sections 2 and 3 contain main theorems. Throughout this paper, fields are all infinite, modules are finite dimensional over these fields. Representation means a matrix representation describing the action of the algebra on a module with respect to a fixed basis.