Is there a highest possible atomic number? I know that atoms with higher atomic numbers tend to become more and more unstable and decay quicker the higher the atomic number goes. Is there a limit to this where the time it takes for the atom to decay is zero? If not, is there some atomic number at which we know that the atom would instantly collapse into a black hole? Are there any other limits that prevent us from going higher, no matter how much energy we put into the atom?
Please excuse my possible layman terms, physics is not my area of expertise.
 A: 
I know that atoms with higher atomic numbers tend to become more and more unstable and decay quicker the higher the atomic number goes.

So far so good, although it is not exactly that atomic number and decay time have such a simple relationship.  Broadly speaking the larger you make a nucleus, the less stable it is likely to be.  However we have no general theory for such complex systems, so there is no defnite rule or law that guarantees that.  Nature might yet surprise us.
The Island of stability referred to by Solomon Slow in a comment is a hypothetical range fo atomic numbers and atomic weights (beyond those we kno) where the stability is hoped to be realtively high.  I don't think anyone expects them to be stable, but just not as unstable as the high atomic number elements we have made to date.  These would still (if they even exist) be very short lived.

Is there a limit to this where the time it takes for the atom to decay is zero?

Not really.  The decays times (which are averages not a fixed time for every atom) will simply get too small to realisticaly say a nucleus really formed.

If not, is there some atomic number at which we know that the atom would instantly collapse into a black hole ? Are there any other limits that prevent us from going higher, no matter how much energy we put into the atom?

Black holes will form if you try and pack too much energy into too small a space.  Essentially there are two mass limits that decide what kind of object acollapse will result in, these are :

*

*the Chandrasekhar limit

*the Tolman-Oppenheimer-Volkhoff limit (the TOV limit)

The Chandrasekhar limit is the maximum mass to form a stable white dwarf.
The TOV limit is the minimum mass to form a black hole.
Between those you get Neutron stars.
So we would need a mass as least as large as the TOV limit to get a black hole, but long before that we would get a rather large star.
There is an outside chance a large nuclear mass much smaller than a star could form a black hole in some unknown way, but to find out if this is possible (or more likely not possible) we would need (at minimum) an advanced theory that incorporated elements of general relativity and quantum theory properly (and we lack such a theory).
