The absolute maximum is the centre of mass energy, 14 TeV. But that's the energy of the colliding protons. In fact, outgoing particles are created through the collision of a constituent of one proton, and the constituent of the other proton. Constituent here means a quark or a gluon. The problem then becomes rather complex, involving what particle physicists call Parton Density Function, i.e. what is the probability to find a given type of parton carrying a given fraction of the proton momentum. But I think you will allow me to give a layman answer and say that introducing a factor 3 (for the 3 quarks the proton is mostly made of) is good enough! So that's a reduction of a factor 10 basically (I mean 9 = 10 at that level of handwaving!).
There is a lot of work on searching for Dark Matter particles at LHC, definitively! A very recent review can be found at https://arxiv.org/abs/1702.02430. This is a "pro" article, but I think you could get some useful info from the conclusion.
A more thorough answer requires to pay attention to those Parton Density Functions (PDF) I mentioned. Let's take an example: it is possible to find a quark up carrying 99% of the proton momentum but this is much less probable than finding one carrying close to 1/3 let's say. So in the absolute, 14 TeV is the maximum but collisions even approaching this would be extremely unlikely. Eventually, it becomes a probabilistic argument because for each maximum mass between 0 and 14 TeV, you have a probability assigned to it. When I wrote 1.4 TeV above, I therefore meant an approximation of the most likely maximum mass, across a vast number of collisions.
From there, if you wanted to see the distribution of probability, it becomes really involved as you have to consider all possible quark-quark, quark-gluon and gluon-gluon collisions. I ought to have the answer as I did my PhD on a very related subject but this was 20 years ago!