Are there viable alternatives to the no boundary proposal? As I understand it, quantum field theory can be described as the evolution of a wave function $\psi_t[\phi]$ depending on some fields, $\phi$. But when we include gravity and we admit that time is measured by clocks and these clocks are themselves made of fields then the wave function cannot depend on an external time, so it is just $\psi[\phi]$. According to the no-boundary proposal this wave function is a functional integral over $\phi'$ of the exponential of some action $e^{-S[\phi']}$ over a Euclidean volume, where the fields have the value of $\phi$ on the boundary. (And perhaps, sums over volumes with different topologies - as expressed in Hawking's black holes and baby universes book).
So far this is very elegant mathematically and conceptually.
Now, this seems to break down with the simple observation that the Universe does not appear to be closed [edit: I mean "compact" as in like a sphere].
Are there any rival theories that are as mathematically elegant but work with a Universe like the one we see?
For example, does combining this idea with string theory, or the holographic principle solve this problem?
 A: I was much confused by this myself, because there is usually no mention of this issue, even in recent publications on the NBWF. I contented myself with the following.
Strictly speaking, what you say is incorrect. The NBWF requires you to exclude non-compact metrics, but not closed does NOT imply non-compact, as is usually erroneously stated. For example a torus is flat but compact. 
Alternatively, you could say that our universe is approximately closed. It is very close to being closed. In its early evolution and at late times it will be approximately de Sitter. In the simplified models, this is enough. 
As to your actual question, the most serious alternative to the NBWF is Vilenkin’s tunnelling proposal. 
Edit: there is another important aspect to consider. There are even no boundary saddle points that have the interior of a Euclidean AdS domain wall going over to an approximate Lorentzian de Sitter regime via a complex transition region. The saddle point geometry itself is not very physical. Once you computed the wavefunction, you use the Cauchy data at the final surface to find the classical evolution using Einstein’s equations.
The no-boundary conditions are usually most important at the “south pole”, at the point where the 4-geometry closes off, they ensure that this happens in a smooth manner. 
