Abstract Suppose ( M n , g ) {(M^{n},g)} is a Riemannian manifold with nonnegative Ricci curvature, and let h d ( M ) {h_{d}(M)} be the dimension of the space of harmonic functions with polynomial growth of growth order at most d. Colding and Minicozzi proved that h d ( M ) {h_{d}(M)} is finite. Later on, there are many researches which give better estimates of h d ( M ) {h_{d}(M)} . In this paper, we study the behavior of h d ( M ) {h_{d}(M)} when d is large. More precisely, suppose ( M n , g ) {(M^{n},g)} has maximal volume growth and has a unique tangent cone at infinity. Then when d is sufficiently large, we obtain some estimates of h d ( M ) {h_{d}(M)} in terms of the growth order d, the dimension n and the asymptotic volume ratio α = lim R → ∞ Vol ( B p ( R ) ) R n {\alpha=\lim_{R\rightarrow\infty}\frac{\mathrm{Vol}(B_{p}(R))}{R^{n}}} . When α = ω n {\alpha=\omega_{n}} , i.e., ( M n , g ) {(M^{n},g)} is isometric to the Euclidean space, the asymptotic behavior obtained in this paper recovers a well-known asymptotic property of h d ( ℝ n ) {h_{d}(\mathbb{R}^{n})} .