Suppose that $X=\{X_t, t\ge 0; \mathbb{P}_μ\}$ is a supercritical superprocess in a locally compact separable metric space $E$. Let $φ_0$ be a positive eigenfunction corresponding to the first eigenvalue $λ_0$ of the generator of the mean semigroup of $X$. Then $M_t:=e^{-λ_0t}\langleφ_0, X_t\rangle$ is a positive martingale. Let $M_\infty$ be the limit of $M_t$. It is known (see, J. Appl. Probab. 46 (2009), 479--496) that $M_\infty$ is non-degenerate iff the $L\log L$ condition is satisfied. In this paper we are mainly interested in the case when the $L\log L$ condition is not satisfied. We prove that, under some conditions, there exist function $γ_t$ on $[0, \infty)$ and a non-degenerate random variable $W$ such that for any finite nonzero Borel measure $μ$ on $E$, $$ \lim_{t\to\infty}γ_t\langle φ_0,X_t\rangle =W,\qquad\mbox{a.s.-}\mathbb{P}_μ. $$ We also give the almost sure limit of $γ_t\langle f,X_t\rangle$ for a class of general test functions $f$.