Is there an upper bound on the gauge group rank in F-theory compactifications on CY 4-folds? It is known that in F-theory compactifications on CY 4-folds one can get gauge groups with very large ranks. The largest single factor* gauge group for compact CY 4-folds I found in the literature is the mind-blowing SO(7232) in this paper: http://arxiv.org/abs/hep-th/9706226 . One can imagine that there exist 4D compactifications of F-theory, which don't have Heterotic duals, with gauge groups that are even bigger in rank. Now, my question is whether there exists an upper bound on the ranks of single factors* in product gauge groups in either F-theory compactifications on CY 4-folds or in G2 holonomy compactifications of M-theory? An order of magnitude answer, if known, would be good enough :) . 
(*) Thanks to @Luboš Motl for correcting my question!
 A: Stringpheno, you are slightly confused about - and you underestimate - the gauge group of the model you linked to. Its rank is not 7232/2, as it would be if the gauge group were SO(7232). The full gauge group of the model actually has rank 302,896 - about 100 times higher than what you say - and SO(7232) is just one of the 251 factors of the gauge group.
I don't remember the exact number but I think that this gauge group is still far from being the world record when it comes to the rank of gauge groups in F-theory geometric engineering, the term you may want to search for if you're interested in these fun constructions. I believe that there are 6-dimensional compactifications with 8 supercharges that admit even simple groups (or at least products of a smaller number of very large factors) as gauge groups which are larger than the product above.
There can clearly exist "infinite" gauge groups but if there is a limit for finite ranks is unknown to me. I believe that these very high gauge groups only appear in F-theory, not in M-theory on G2 manifolds, and it's arguably not just due to the limited creativity of the geometric engineers.
