In ordinary representation theory over $\mathbb{C}$, all the irreducible modules of a finite group $G$ appear as composition factors of the tensor products $X \otimes \cdots \otimes X$ of a faithful $\mathbb{C}$-representation $X$.
This is e.g. Thm. 10.8 in Ch.V of Huppert's Book "Finite Groups I".


This should also be true in positive characteristic,
so let $X:G \to \mathrm{GL}(n,K)$ with $\mathrm{char}(K) = p$ be a modular faithful representation. Then all irreducible $KG$-modules appear as composition factors of the tensor products $X \otimes \cdots \otimes X$. 


I'm not sure, how to prove this, but I think it shouldn't be difficult.



Composition factors of tensor products of modular representations
This is true. I believe it may be due to L.G. Kovacs. When $K$ is algebraically closed, the proof is much the same as the complex case, except that one works with the Brauer character $\phi$ afforded by $X.$ For any Brauer irreducible character $\psi,$ let $\theta$ denote the Brauer character of its projective cover. Then some linear combination of powers of $\phi$ vanishes on all non-identity $p$-regular elements, but not on $1\_{G}.$ Hence $\langle \theta, \phi^{n} \rangle \neq 0$ for some $n,$ so that the simple module affording $\psi$ is a composition factor of $X \otimes \ldots \otimes X$ ( $n$ factors) The case when $K$ is a finite field follows from the algebraically closed case.


