Is the twisted symmetric fifth power $L$-function holomorphic? Let $\pi$ be a Maass cusp form for SL($2,\mathbb Z$). Let $\omega$ be a primitive Dirichlet character. 
Let us consider the $L-$ function
$$L(s,Sym^5 \pi \times \omega)$$ or $L(s,Sym^6 \pi \times \omega)$ or $L(s,Sym^7 \pi \times \omega)$ or $L(s,Sym^8 \pi \times \omega)$.
Are they known to be holomorphic on the whole complex plane?
 A: This is definitely an open problem.
For a very recent reference of $Sym^m\pi$ being known only for $m \leq 4$ ($\pi$ arbitrary cuspidal), see:


*

*Huixue Lao, Mark McKee, Yangbo Ye, Asymptotics for cuspidal representations by functoriality from $GL(2)$ (2015)


As eric mentions in the comments, arithmetic techniques make some cases like holomorphic forms accesible, but those techniques are certenly missing in this case, $\pi$ a Maass form.
Also, more concretely for Maass forms, see this paragraph on a 2003 Sarnak paper (Spectra of Hyperbolic Surfaces):

This [Kim-Sarnak result towards the eigenvalue conjecture for Maass
  forms] is getting close to $1/4$, but it is also close to the limit of
  these methods. The functorial lifts $sym^3$ and $sym^4$ are based on
  the continuous spectrum (Eisenstein series) on exceptional groups
  including $E_8$. What can be done this way terminates with the finite
  list of exceptional groups.

This is closely related with what Marty and Garret said in their very interesting comments above.
