There's a famous theorem of Serre that if $E$ is a non-CM elliptic curve over $\mathbf{Q}$, and $\rho_{E, \ell} : Gal(\overline{\mathbf{Q}}/{\mathbf{Q}}) \to GL_2(\mathbf{Z}_\ell)$ is its $\ell$-adic Galois representation, then the product $\rho = \prod_{\ell} \rho_\ell: Gal(\overline{\mathbf{Q}}/{\mathbf{Q}}) \to GL_2(\widehat{\mathbf{Z}})$ has open image. In particular, this implies that $\rho_{E, \ell}$ is surjective for almost all $\ell$; but is much stronger than this, as it shows that the $\rho_{E, \ell}$ for different $\ell$ are "independent" in some sense.

If one works instead with general non-CM modular forms $f$ of weight $k \ge 2$, then I know of theorems (due to Ribet and Momose) describing the images of the $\rho_{f, \ell}$, showing that they are "as large as possible" for almost all $\ell$ (cf. this earlier question of mine). (The notion of "as large as possible" is much more delicate in this generality, because the coefficient field can be nontrivial, and there can be "inner twists".)

Are there analogues of Serre's adelic open image results for higher-weight modular forms?

EDIT. I'll just put up a *guess* of mine, just to show that there is a reasonable conjectural formulation which is compatible with the known results for individual $\ell$. Momose has shown that there is a subfield $F$ of the coefficient field $E = \mathbf{Q}(f)$, a quaternion algebra $B$ over $F$, and an open subgroup $H$ of $Gal(\overline{\mathbf{Q}}/{\mathbf{Q}})$, all independent of $\ell$, such that for any $\ell$ the representation $\rho_{f, \ell}: Gal(\overline{\mathbf{Q}}/{\mathbf{Q}}) \to GL_2(E \otimes \mathbf{Q}_\ell)$ sends $H$ to an open subgroup of the group $\{ x \in B(F \otimes \mathbf{Q}_{\ell}) : \operatorname{norm}(x) \in \mathbf{Z}_{\ell}^{\times(k-1)}\}$. For all $\ell$ coprime to the discriminant of $B$, we have $B(F \otimes \mathbf{Q}_{\ell}) = GL_2(F \otimes \mathbf{Q}_{\ell})$, and Ribet has shown that for all but finitely many such $\ell$ the image of $H$ is all of $\{ x \in GL_2(O_F \otimes \mathbf{Z}_{\ell}) : \operatorname{det}(x) \in \mathbf{Z}_{\ell}^{\times(k-1)}\}$.

Conjecture: The image of $H$ in $GL_2(L \otimes \mathbf{A})$ contains an open subgroup of $\{ x \in B(O_F \otimes \widehat{\mathbf{Z}}) : \operatorname{norm}(x) \in \widehat{\mathbf{Z}}^{\times(k-1)}\}$.

This is visibly consistent with (and implies) Momose and Ribet's results.

conjectured, at least. The standard reference is Serre's "Proprietes conjecturales des groups de Galois Motiviques..." in volume 1 of the book Motives, PSPM 55. Specifically, section 11 is about your problem in the more general setting of systems of representations attached to motives, where conjectural statement 11.4? page 390 is a very general conjecture which should specialize to yours in the case of a modular form (I have not checked). Now, as for the proof, I don't know. Serre might say something about the status on page 392 which I can't access now. $\endgroup$ – Joël Sep 23 '14 at 15:41