Does a quantum wave stretch out forever? In learning about the duality of quantum particles, I wonder if a quantum wave stretches out into the distance, essentially forever?  And if so, when a particle is observed, is it possible (but highly unlikely) that the particle might be found anywhere along the wave?
 A: 
In learning about the duality of quantum particles, I wonder if a
quantum wave stretches out into the distance, essentially forever?

Not necessarily although in many cases: yes.
Consider the educational case of the particle in a 1D box (or a 2D or 3D box, for that matter)
Here the particle is strictly confined to the domain $[0,L]$ with $\text{zero}$ probability of finding it outside of these bounds.
For atoms however the probability density function of electrons, $|\Psi|^2$, decays exponentially with increasing distance (from the nucleus) and only tends $\to 0$ for infinite distance. In reality however the probability tails off pretty quickly.

Finally a word of caution against the use of the term "quantum wave". In QP, particles are assigned a so-called wave function, $\Psi(r)$, found by solving the Schrödinger equation of the system. The wave function shouldn't be considered a wave in the strict sense of the word. Rather, the square of its modulus, $|\Psi(r)|^2$, gives the probability density function of the particle.
The wave function itself is a mathematical object that contains all the information about the quantum system that is knowable.
A: Nothing that starts out in a finite region later extends to an infinite region, because it would take an infinite amount of time to get there. This is true of wavefunctions in quantum physics just as much as it is true of more familiar things. But we often use infinitely extended wavefunctions for purposes of analysis, a bit like using perfect waveforms in Fourier analysis.
Someone will say that a wavefunction such as the ground state of the electron in hydrogen decays exponentially but extends to infinity in principle. But no electron ever gets exactly into the ground state because it would take an infinite amount of time. So if we want to talk about exact statements then, once again, no wavefunction extends to infinity.
Having said that, I would be happy to say that some given hydrogen atom is genuinely in its ground state after say 100 decay lifetimes, because that is consistent with all feasible observations. Does this mean I would be willing to admit the wavefunction now extends to infinity? Well not really. Even if a wavefunction has the form $\exp(-r/a)$ with $a$ a constant and $r$ unbounded, then to say that it extends to infinity is to miss the essential physics. When $r > 1000 a$ in this example, the wavefunction approaches so closely to zero that no theory in physics is accurate enough to say whether or not it is truly zero.
Finally, the momentum eigenstate wavefunctions that are often used in analysis are useful mathematical quantities, but no particle ever manages to end up in an exact momentum eigenstate.
If you want an example of a really big wavefunction, then I think there may be some examples in astrophysics, when light, and also neutrinos and gravitational waves, propagate over very large distances. If it can be modeled as a spherical wave and has not undergone scattering so as to pick one direction over another, then the spherical wavefunction can in principle extend over many lightyears after the light (or neutrino etc.) has journeyed for long enough.
