Occurrence of the trivial representation in restrictions of Lie group representations

The short answer is: no. In theory, there are explicit formulae for branchings based on the Weyl character formula, but no reasonable person would call these simple.

There are interesting results which give you some information about branching multiplicities, but all involve real work. For instance, if G and H are compact, you can obtain asymptotic information about H-invariants in the representations $V_{n\lambda}$ as a function of $n$: it's a polynomial, whose leading order is the dimension of $\mathcal{O}_{\lambda}//H $, the symplectic reduction of the coadjoint orbit through $\lambda$ by $H$, and whose leading coefficient is the symplectic volume of this manifold.

If you prefer algebraic geometry, this polynomial is the Hilbert polynomial of the corresponding GIT quotient.

If $H$ is a root subalgebra, then life is a bit easier, and you can use combinatorial methods like crystals, but this is still not easy.

Consider the case that $H$ is a maximal torus of $G'$, and your $G = G' \times H$. (Well, you said $G,H$ semisimple, but I'm going to pretend you meant reductive, because really you should have.) Then your question is answered by the Kostant multiplicity formula.

If you're willing to take that formula as "simple", then yes, the general case is not much harder. Let $T_H$ be a maximal torus for $H$, and $T_G$ a maximal torus that contains it. Use the Kostant multiplicity formula to go from $G$ to $T_G$; now we have a function on the weight lattice $T_G^*$ of $T_G$. Push it forward under the restriction map $T_G^* \to T_H^*$. Difference that in the directions of all the positive roots of $H$. Look at the value at $0$.

If you want a positive formula (like I do!) then none is known. What you're asking for includes the case that $G = H\times H\times H$, and then the question becomes one about computing tensor product decompositions. That subcase does have positive formulae, e.g. counting Littelmann paths, but noone has extended it even to the case of branching from $H \times K$ to $H$ where $K$ is a symmetric subgroup of $H$.

I would like to add to the answers by Ben and Allen. First if we extend the question to include all multiplicities and not just the multiplicity of the trivial representation then there are a number of special cases that are of interest:

1.Take $H$ to be the trivial group then the question asks for the dimension of a representation.
2.Take $H$ to be a maximal torus then we are asking for the character of a representation.
3. Take $G=H\times H$ and $H$ the diagonal subgroup. Then we are asking for tensor product multiplicities. 4. For $V$ a representation of $K$. Take $G=SL(V)$ and $H=K$. Then we are calculating plethysms.

A paper that discusses this which gives a formula for branching rules is:

MR1120029 (92f:22022) Cohen, Arjeh M. ; Ruitenburg, G. C. M. Generating functions and Lie groups. Computational aspects of Lie group representations and related topics (Amsterdam, 1990), 19--28, CWI Tract, 84, Math. Centrum, Centrum Wisk. Inform., Amsterdam, 1991.

As I understand it both Ben and Allen agree that this is not a simple way of finding branching rules. The reason is that this involves a sum over the Weyl group.

If you take the special cases above then historically the first solutions to these problems were given by formulae involving a sum over the Weyl group. For some of these special cases there are solutions which don't involve cancelling terms. For example, LiE calculates these without summing over the Weyl group. The LiE home page is
http://www-math.univ-poitiers.fr/~maavl/LiE/index.html
and the LiE manual does describe how these special cases are implemented.

However LiE treats each of these special cases separately. I think it is an interesting question whether there is an algorithm for finding branching rules which could be implemented in LiE and which does not involve a sum over the Weyl group.