Sequential Double Double Slit Experiment? Let's say you arrange the double slit experiment so that there are two sets of two slits one after the other. Like so:
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If you measure which slit the photon went through in the first set, will the wave function remain collapsed as it passes through the second set so that you get two points on the detector screen, and not an interference pattern?
 A: Depends what you mean by "measuring". Obviously, if the photon is absorbed, then nothing at all will happen afterwards, as pointed out in the other answer.
You can however also imagine a way to know which slit the photon went through without absorbing it. For example, you can make it so that when the photon passes through the upper slit its polarization is $H$, while when it goes through the lower slit it's $V$ (assuming the polarization doesn't change in the rest of the evolution). Then by measuring the polarization at the screen at the end you are effectively measuring which slit the photon went through in the first stage. This is just an example of how you could do this. An ever simpler scenario is that you just block one of the initial slits, which then gives you the same identical evolution as if you measured the photon to have passed through the other slit in some other way.
Regardless of how you achieve this, assuming for example the photon was measured to pass through the upper slit in the first stage, then everything evolves from there as if you just started the photon from there. You thus get an interference pattern due to the second set of slits, as if you just performed a standard single double-slit experiment.
A: If you measure a photon then it is fully absorbed and the wave function is collapsed and it does not go onwards at all.  Physicists try use special materials that can make a higher energy photon produce 2 lower energy photons that are entangled, and these 2 photons will proceed onwards (the first photon is collapsed).  The 2 photons will proceed onwards in specific directions and physicists will try and detect the position of one photon so that they know the position (slit) of the second one but when this happens they find no interference.
Possibly when you add extra slits to the second photon you do create the uncertainty again and therefore interference should be possible further along.
