  We study F-theory compactified on elliptic Calabi-Yau threefolds that are
realised as hypersurfaces in toric varieties. The enhanced gauge group as well
as the number of massless tensor multiplets has a very simple description in
terms of toric geometry. We find a large number of examples where the gauge
group is not a subgroup of E8xE8, but rather, is much bigger (with rank as high
as 296). The largest of these groups is the group recently found by Aspinwall
and Gross. Our algorithm can also be applied to elliptic fourfolds, for which
the groups can become extremely large indeed (with rank as high as 121328). We
present the gauge content for two of the fourfolds recently studied by Klemm et
al.
