How can I tell how many power supply's it is safe to put in series?
It depends on the isolation potential each supply is capable of.  That's the maximum allowed voltage difference from any output to any input.  For most commercial power supplies, that's often in the 1.5 to 2.5 kV range.

Let's say you have a bunch of 15 V supplies running from 120 VAC, and they are rated for 2 kV isolation.  Since the input is a sine, it will range from &plusmn;170 V from ground, so let's use 200 V.  If one of the outputs is grounded, then we have up to 215 V difference from any output to any input.

From the above, there is (2 kV)-(215 V) = 1785 V of isolation potential left.  For each supply from the one that is grounded, another 15 V is used up.  That comes out to 119 supplies pn either side of one that is grounded.  Since two supplies in the middle can share a ground connection, the absolute maximum in this example just from isolation voltage is 119 + 2 + 119 = 240.

So basically, the answer is "a lot".  Note that in the maximum string of supplies above, the total end to end voltage would be 3.6 kV.  You'd of course have to be very careful with that, and make sure any insulation you are using can handle that.  Ordinary off the shelf wire doesn't usually have 3.6 kV insulation rating.

I also want to make it clear I'm not actually recommending or endorsing this.  This is a bad idea.  One problem is that the isolation from output to case is probably less than output to input.  This means that effectively the cases of the string of power supplies must be considered "live".  You can't, for example, have the string of 240 supplies sitting on a metal rack.  All but a few of the supplies in the middle would be unsafe to touch.  The math above also doesn't leave any margin for voltage spikes on the power line and the like.  Again, more than "a few" supplies in series is a bad idea.