Is there an example of a situation where you need a continuous spectrum? If you had a hydrogen atom you could say that you want to be able to ionize them. But if you then add the potential due to the earth, e.g. $$V=\frac{-Gm_eM_\oplus}{\sqrt{(x_e-x_\oplus)^2+(y_e-y_\oplus)^2+(z_e-z_\oplus)^2}}$$ where $(x_\oplus,y_\oplus,z_\oplus)$ is the center of the earth, and $(x_e,y_e,z_e)$ is the location of the electron.
Now electrons are bound unless they achieve earth escape velocity. And if that isn't good enough you can add the potential $$V=\frac{-Gm_eM_\odot}{\sqrt{(x_e-x_\odot)^2+(y_e-y_\odot)^2+(z_e-z_\odot)^2}}$$ where $(x_\odot,y_\odot,z_\odot)$ is the center of the sun.
Now electrons are bound unless they achieve solar system escape velocity. And if that isn't good enough you can add an external potential for the mass in the galaxy. And that's good unless they have enough energy to escape the galaxy.
And you can argue that that is less than an eV of ionization so we need more energy but you could always put the whole galaxy in orbit around some super cluster far away with a huge escape velocity (we could even make the escape velocity be $c$ so it works for any non relativistic particle) and it seems unlikely to affect everything over here very much when we are a nice high quantum number angular state going about that super far away supercluster.
Not that I'm not saying that we are orbiting a giant supercluster I'm saying that it doesn't seem like it has immediate experimental consequences for most situations. So it seems like it doesn't matter much whether we are bound or are unbound.
Is there is a situation where we need unbound states rather than it just being convenient? I'm not asking for a list of them, just whether there are situations where we need them.
 A: For example, the escape velocity of a particle from the galaxy is about 400 km/s and in most conceivable circumstances (unless you are basically on top of the event horizon of a black hole or on the surface of a neutron star), escape velocities will be far, far below relativistic speeds (here defined as $3\times10^4$ km/s). So basically, if a particle has a relativistic speed it is almost certainly unbound.
Moreover, if you were to compute the "bound" eigenstates corresponding to an electron in the potential generated by our galaxy (for example) you would find that they were essentially identical (to an absolutely excellent approximation) to the eigenstates of a free particle.
The spectrum would be essentially continuous as can be seen in the expression for the energy spacing of a particle in a box (of length $L$):
$E = \frac{\hbar^2 k^2}{2 m}$
with allowed $ k $ being:
$ k = \frac{n \pi}{L} $
so that the energy states are:
$ E = \frac{\hbar n^2 \pi^2}{2mL^2}$
so when we take $L\rightarrow \infty$ (which we essentially do in the case of the box being the size of a galaxy), we recover the free particle case!
A: I am puzzled at your leap: quantum bound state of electron to hydrogen, to the earth potential classical bound state. Bound classically and bound quantum mechanically are two different frameworks.
The electron is bound quantum mechanically  to the hydrogen atom and does not "see" the gravitational coupling quantum mechanically due to the very small value of the gravitational coupling constant with respect to electromagnetic one that enters in the quantum calculation (proton electron interaction). Similar argument as that space expansion does not expand atoms. 
The classical gravitational attraction of an electron to the mass of the earth will not give a quantum mechanical orbital (bound state with the center of the earth), again because of the coupling constants. If in vacuum the electron could have a classical orbit unless it had escape velocity.
A: One usually goes to the continuum because of its nice mathematical properties; lattice QFT is a hint of how hard a quantum theory becomes if we break the symmetries. For scattering theory, you usually want to be able to apply Lorentz invariance, or its classical counterpart, Galilean, and this already implies you are working with the continuum spectra of momentum. And remember that you need Lorentz group to classify spin.
Besides symmetries, another motivation to teach the continuum specta is because there are states with the same energy that some of the states of the continuum but very different properties. Search google for "states insde the continuum spectrum" and a lot of content in book, both in theoretical and applied physics, will surface. Randomly from the search, let me cite the abstract of this article: 

Quantum mechanics predicts that certain stationary potentials can
  sustain bound states with an energy buried in the continuous spectrum
  of scattered states, the so-called bound states in the continuum
  (BIC). Originally regarded as mathematical curiosities, BIC have found
  an increasing interest in recent years, particularly in quantum and
  classical transport of matter and optical waves in mesoscopic and
  photonic systems

Of course you can still have the cake if you regularize the continuum by a very slowly decreasing potential as suggested (box is definitely bad idea, because it breaks also some discrete states of 1/r potentials) but instead of working with two different kinds of discrete sets, it is more clear to leave the continuum to be continuum and not some obscure subset of the rational numbers.
Last consideration is that we want to keep regularisation and cut-offs as a last-resource weapon. Divergences in advanced models can force us to use them, and calling for such tools before the need arises can be confusing to the student.  
