Are wave functions built of other wave functions? I have heard the term wave function used in a number of contexts describing both quantum objects (electrons, etc) and more macroscopic objects (atoms, birds, cats, etc). Is it correct to say a lithium atom, for example, has a wave function that can be described as an entangled set of wave functions that describe the electrons, protons, etc that make up the atom? If so are we able to extend this to talk about the wave function of a bird or even the whole universe?
 A: Take a Hydrogen atom (simpler than a Lithium one) made of just one electron and one proton.
The complete orthonormal bases that span the Hilbert for the electron and for the proton are $\{|\phi_{\mathrm{el}}^i\rangle$} and $\{|\phi_{\mathrm{nuc}}^j\rangle\}$ respectively. 
The state of the atom $|\Psi\rangle$ can then be expressed as 
$$ |\Psi\rangle = \sum_{ij} c_{ij}|\phi_{\mathrm{el}}^i\rangle \otimes \phi_{\mathrm{nuc}}^j\rangle, $$
i.e. the basis for the whole atom is the tensor product of the bases of the inner constituents.
In general, $c_{ij} \neq c_i c_j$, i.e. you cannot decouple the electron from the nucleus and you have to stick to an entangled wavefunction, where the behaviour of one is affected by the behaviour of the other.
If the inner constituents of the atom can be treated separately (e.g. under the Born-Oppenheimer approximation), then you can simplify the problem by solving two Schrödinger equations, two Hamiltonians etc.
In this case, the electron and the proton are disentangled, $c_{ij} = c_i c_j$ and so:
$$ |\Psi\rangle = \sum_{ij} c_{i} c_j|\phi_{\mathrm{el}}^i\rangle \otimes |\phi_{\mathrm{nuc}}^j\rangle = \sum_{i} c_{i} |\phi_{\mathrm{el}}^i\rangle \otimes \sum_j c_j |\phi_{\mathrm{nuc}}^j\rangle = |\Phi_{\mathrm{el}}\rangle \otimes \Phi_{\mathrm{nuc}}\rangle, $$
i.e. you can constuct the total state directly from the states of the constituents
This reasoning then extends to other atoms, and, eventually, to birds and cats. At least on paper.
EDIT 1: for macroscopic, non isolated objects, decoherence and coupling to the environment causes quantum effects to wash out. You can still have macroscopic quantum objects, like superfluids, if carefully prepared and controlled). 
EDIT 2: so if every single constituent of the universe were not interacting with anything else , then you could trivially construct the wavefunction of the whole universe just by the tensor product of the individual wavefunctions.
Of course stuff does interact so you can’t do this in practice.
The “wavefunction of the whole universe” is also (and maybe better) discussed in terms of density matrices, and pure and mixed states. I invite you to have a look at that stuff too.
A: I don't think this question has really anything to do with a specific physical system.
A "wavefunction" is really just one way to describe a state, in which we characterise the state via the set of amplitudes in some continuous basis (such as the position basis). However, there isn't really anything special in this choice of basis, from a fundamental point of view. Because of this, you can forget about wavefunctions and just think more generally about quantum state decomposed in some arbitrary basis.
Using bra-ket notation, we can write a generic state as something of the form $|\psi\rangle=\sum_k c_k|k\rangle$. Replace the sum with an integral where preferred. So can such a state be thought of as "an entangled set" of other states?
Well, sort of, once we realise that "entanglement" is not really the right terminology here.
To talk of "entanglement", one needs a partite system of some sort. For a state such as the above $|\psi\rangle$, we don't have this. However, it is definitely true that you can always write a state $|\psi\rangle$ as a coherent superposition of other states, and you can do this in an infinite number of ways (just as you can write a vector as sum of other vectors in an infinite number of ways).
Now, if $|\psi\rangle$ is a partite system, e.g. something of the form $|\psi\rangle=\sum_k c_{ij} |\psi_i\rangle\otimes|\phi_j\rangle$, then it is true that any such state can be written as a coherent superposition of other entangled states.

If so are we able to extend this to talk about the wave function of a bird or even the whole universe?

This just looks like a non sequitur to me.
