The present paper is concerned with functions analytic in the unit disc having rapidly increasing maximum moduli. To study the precise rates of growth of such functions the concept of index is introduced. Several growth parameters in terms of the index are defined for a function analytic in the unit disc and their characterizations in terms of the Taylorseries development of the function are obtained. The results in the present paper improve and refine the earlier results ofSons (J. Math. Anal. Appl,24 (1968), pp. 296–306),MacLane (Asymptotic values of [holomorphic functions,Rice University Studies, Houston, 1963), andKapoor andJuneja (Indian J. Pura Appl. Kath.,7(3) (1976), pp. 241–248).