Which nucleus is the most resilient against gamma-induced fission? To state the title question perhaps more precisely:
What is the largest photon energy $E_{\gamma}$ and the corresponding mass number $A$ and atomic number $Z$ of a suitable nucleus ${}^A_ZX$ (presumably in a ground state) such that the hypothetical reaction
$$ {}^A_ZX + \gamma \rightarrow {}^{(A - a)}_{(Z - z)}Y + \text{whatever remains (with combined charge} +z \text{)}$$
is "kinematically" forbidden for any values $1 \le a < A$ and $Z \ge z \ge Z + a - A$,
while conforming to the standard model?
Edit
Changed the question title (removed the parenthetical qualification "whether otherwise stable or not"): for any unstable nucleus the stated question and condition is not meaningful and not relevant.  
 A: Nuclear physics is at the realm of quantum mechanics, fission and fusion are reached when the  quantum mechanical solution for the reaction nucleus+photon--> A+B+C ....  fragment nuclei gives a measurable probability. It will be a different problem for different nuclei that will require a quantum mechanical model for its solution.
The process of fission by gammas is called photofission and is used for detecting fissionable materials. . The gamma ray energies used are :

Since the photofission cross section for most fissionable materials drops to near zero for incident photon energies of less than 6 MeV, a source of photons with a higher energy is needed, for example 9 MeV.

So the order of magnitude is close to the nickel binding energy per nucleon given by Chris, but this is fissionable material.  I have not been able to find photofission crossections for all the periodic table, as the research interest has been to use photons instead of neutron in inducing fission in already known fissionable  materials, so this is only a partial answer.
A: The atom with the highest binding energy per nuclei is Nickel-62 with a binding energy of 8.7945MeV. As there exist photons at that energy, it is theoretically possible to break the binding energy of Nickel-62 with a photon, thus rendering it two smaller atoms. I cant find any evidence that this has been done before though. As there exist no atoms with a greater binding energy, this is the theoretical maximum photon energy.
