This paper is devoted to proving some asymptotic regularity of the solutions of the p-Laplacian equation ut−div(|∇u|p−2∇u)+f(u)=g(x) (p∊(2,N)) considered on a bounded domain Ω⊂RN(N≥3). The nonlinear term f satisfies the polynomial growth condition of arbitrary order c1|u|q−k≤f(u)u≤c2|u|q+k, where q≥2 is arbitrary. As an application of the asymptotic regularity results, we not only can obtain the existence of a (L2(Ω),W01,p(Ω)∩Lq(Ω))-global attractor A immediately but also can show further that A attracts every bounded subsets of L2(Ω) under the W01,p∩Lq+δ-norm for any δ∊[0,∞). Furthermore, the fractal dimension of A is finite in Lq+δ(Ω) for any δ∊[0,∞).