Calling '$L$function' any automorphic $L$function belonging to the Selberg class, what are the known $L$functions $L(s,F)$ and $L(s,G)$ of respective degrees $d$ and $d'$ such that the RankinSelberg convolution $L(s, F \otimes G)$ is provably (as of today, December 13 2016) an $L$function of degree $dd'$?
An automorphic Lfunction $L(\pi,s)$ belongs to the Selberg class if and only if the generalized Ramanujan conjecture for $\pi$ is known.
This includes two important cases: Hecke characters and holomorphic modular forms.
Automorphicity of RankinSelberg convolution is known in that range for $\mathrm{GL}(1)\times \mathrm{GL}(1)$ and $\mathrm{GL}(1)\times \mathrm{GL}(2)$ (classical) and $\mathrm{GL}(2)\times \mathrm{GL}(2)$ (Ramakrishnan).
Therefore the answer includes Hecke Lfunctions and Lfunctions of modular forms.
I'm not sure if there are any more cases where both neccesary results are known. For example for Maass forms the Ramanujan conjecture is missing. See the comments for more information.

1$\begingroup$ I am fairly sure that the generalised Ramanujan conjecture is known in a bit more generality than that! See e.g. Clozel's article "Purity Reigns Supreme", imjprg.fr/fa/bpFiles/Clozel2.pdf, which shows for instance that a selfdual cohomological cuspidal automorphic representation of $GL_n / \mathbf{Q}$ of conductor 1 must satisfy Ramanujan. $\endgroup$ – David Loeffler Dec 14 '16 at 9:46

$\begingroup$ @David Loeffler : would preprints.ihes.fr/2016/M/M1605.pdf provide other examples of Lfunctions satisfying the required property ? $\endgroup$ – Sylvain JULIEN Dec 14 '16 at 10:10

$\begingroup$ @DavidLoeffler You're right, I didn't mean to say that those were all the known cases. I'll fix it. Do you know of other cases where both Ramanujan and modularity of RS are known? $\endgroup$ – Myshkin Dec 14 '16 at 11:47

1$\begingroup$ What are you saying about GL(2) x GL(3)? Functoriality is known by KimShahidi. $\endgroup$ – Kimball Dec 14 '16 at 14:55

$\begingroup$ I'd say It also includes the RankinSelberg convolution of those $\endgroup$ – reuns Dec 14 '16 at 15:30