TITLE: Are there "unsociable" irreps? (Definition inside)
QUESTION [9 upvotes]: If, with a bit abuse of notation, $U\notin R\bigotimes{R}\bigotimes{R}\bigotimes{...}$ (i.e., regardless how long you clebsch up $R$, $U$ won't appear in the expansion), I call $U$ unsociable with respect to $R$ (in the group $G$). Clearly everything else is unsociable with the $1$ irrep, since (duh) $1\bigotimes{1}\bigotimes...=1$. For finite groups there can be nontrivial (ahem) cases: For $G=C_{3v}$ (the symmetry group), $A_2\bigotimes{A_2}=A_1$ and thus $E$ is unsociable w.r.t. $A_2$.
But what is the deal with Lie groups? E.g. is the defining irrep of $E_7$ unsociable w.r.t. the adjoint one?

REPLY [10 votes]: This is meant as an extended comment (sometimes correction) to things said in various answers and comments.   As a public service I'll try to fill in some of the history and references.  I recall some of this coming up previously on MO (exercise: find it).
In the framework of representation (or character) theory of a finite group $G$ over $\mathbb{C}$, Burnside led the way in his 1911 treatise (Chap. XV, Thm. IV): given a faithful representation of $G$, all irreducible representations (or characters) occur in some tensor power.    A proof of this is given in the 1962 Curtis-Reiner book (32.9); a later addendum to that book refers also to the slightly later work of Brauer and his student Steinberg.   Note that faithfulness is an obvious necessary condition here, not emphasized in most sources.   
Steinberg, Complete sets of representations of algebras, Proc. Amer. Math. Soc. 13 (1962), 746-747, offered a simpler proof of Burnside's theorem.  Then Brauer, A note on theorems of Burnside and Blichfeldt, Proc. Amer. Math. Soc. 15 (1964), 31-34, provided his own version of Burnside's theorem with a number of nice refinements: limiting how many tensor powers are needed, limiting the size of the characteristic 0 field, commenting on what remains true in prime characteristic.   Both of these papers are available online at the AMS website. 
Meanwhile, in his 1946 book Theory of Lie Groups (Chapter VI), Chevalley treated compact Lie groups and essentially provided a variant of Burnside's theorem.   But here it's essential to use both the faithful representation and its contragredient (dual).   The proof is in the context of the ring of representative functions, Tannaka duality, Peter-Weyl theorem.   
In a later short note, G.I. Lehrer provided a much more algebraic proof of Chevalley's theorem (again for compact Lie groups):  Produits tensoriels de grepresentations de groupes de Lie, C.R. Acad. Sci. Paris 275 (1972).   He also refers to Steinberg's paper.
From compact Lie groups it's not so far to the finite dimensional representation theory of a semisimple Lie algebra or algebraic group in characteristic 0.  But Burnside's theorem and its variants take their sharpest form for groups, whereas a single Lie algebra may belong to simple Lie or algebraic groups or the same Lie type but different isogeny type.    This conceals the requirement that one start with a faithful representation.