In school, I am learning about the Double Slit experiment and it's implications on Quantum Mechanics. I was wondering if the double slit experiment was possible with many different slits and screens. For example, if there were 5 sets of slits, light would pass through the first set of slits, and go into a second set of slits, which would be angled at 45˚(or some arbitrary angle) to the first one. If this process is repeated 5 times in total, would there still be some interference effect observed on the final screen.
A double slit is usually in a first approximation (which you are dealing with in school, probably) just a pair of point sources which are a distance $d$ apart. Sure, each of the slits has technically a width $w$, but unless you are looking at double-slit diffraction and not the simpler$^1$ double-slit interference, you don't care about the size of the individual slits.
Now, the Huygens-Fresnel principle states that in the double-slit experiment, both slits act as point sources emitting spherical waves which will interfere everywhere on the other side on the slit screen. The detection screen is just there to allow for a simple geometric calculation of the pattern along that particular screen.
If you put a second slit screen behind the first one, possibly angled at an angle, no matter what kind of interference pattern might show up along that second slit-screen, the slits will still be pointlike and the secondary (and all subsequent after that) interference pattern will be more or less the same as the first one, barring intensity losses.
So in the first order approximation used in school and introductory optics classes, the interference pattern will not look any different.
Now, of course, if you extend your calculations to account for the diffraction each of the slits composing the double slit, their width $w$ comes into play and since the Huygens-Fresnel principle still says that each point in the slit emits a spherical wave, you will have to take into account the intensity variations the first screen will cause on subsequent ones.
$^1$ Interference and diffraction are more or less the same thing in the sense that both are well explained with the Huygens-Fresnel principle. Usually in school, one deals only with double-slit interference, but you can of course ask what happens to the single-slit properties of each of the slits and how they effect the overall interference picture. That runs under the name of double-slit diffraction, see e.g. in these great MIT notes
The most common way to work out the solution for this kind of problem is by using the wave function.
It works like this: Before the particle is observed (the observation is in this case the particle hitting the screen) its motion is described by a wave.
The wave has a "wavy" dinamics described by Shroedinger's equation which can be completely solved, in principle, in any situation. The wave passes through any kind of experiment you like, maybe a lot of slits, as you said, and than, when we want to "observe" the particle, we make the wave hit a screen.
Finally, the amplitude of the wave at the screen will give us the probability of the particle hitting each part of the screen.
The problem here is determining the way the wave function passes through your experiment. But an interference pattern is still going to happen in some way.
Maybe if you have a complicated set of slits, with angles between then and so on... the final pattern won't be simple or "beautiful" to see... as you would expect from an experiment made with water waves, for exemple, where they pass through many sets of slits.