5, but it also depends on the circumstances. What liquid is used, temperature, viscosity, etc. There's some material science stuff that's far beyond the intended scope of this question.
Unless I completely misunderstand how this works, I think 5 is the only one that will fill up. It then overflows, preventing any of the taller ones from filling. 7 is shallower but won't start filling until 3 gets fuller than 5, which it never will. I assume the blockage between 2 and 3 is a mistake.
5 will be the only one that will ever fill up unless you really crank up the pressure in which case 7 will also fill but very slowly. 5 and 7 are open containers and there's a hole in the bottom of 4. But if it's water coming out of a tap then only 5 will fill .
Wouldn't scale and viscosity play a role? Seriously, imagine a river vs a capillary tube. Also how many dimensions? And forces involved? Is that a blockage between 2 and 3? Are the walls breakable? How will the fluid hold air? Are the lines into structure 5 lower than the walls? Is this in a vacuum?
I got it! First, the free floating faucet will drop into bucket one. The impact will certainly break its connecting tube and broken 1 + faucet collapse into 4. Therefore 4 will be broken but full of shards.
Vessel 5 is the lowest point, so it might seem like it would fill first.
However, if the faucet pours water faster than it can flow through the tube from Vessel 1 to Vessel 2, and as the faucet is depicted as having a larger opening, Vessel 1 will fill up first because the water can't flow out quickly enough to the other vessels.
The number of people in these comments who already understand the self-siphoning nature of water with zero explanation required makes me so proud to be here among them.