We consider a new large-N limit, in which the t Hooft coupling grows with N. We argue that a class of large-N equivalences, which is known to hold in the t Hooft limit, can be extended to this very strongly coupled limit. Hence this limit may lead to a new way of studying corrections to the t Hooft limit, while keeping nice properties of the latter. As a concrete example, we describe large-N equivalences between the ABJM theory and its orientifold projection. The equivalence implies that operators neutral under the projection symmetry have the same correlation functions in two theories at large-N. Usual field theory arguments are valid when t Hooft coupling $lambdasim N/k$ is fixed and observables can be computed by using a planar diagrammatic expansion. With the help of the AdS/CFT correspondence, we argue that the equivalence extends to stronger coupling regions, $Ngg k$, including the M-theory region $Ngg k^5$. We further argue that the orbifold/orientifold equivalences between certain Yang-Mills theories can also be generalized. Such equivalences can be tested both analytically and numerically. Based on calculations of the free energy, we conjecture that the equivalences hold because planar dominance persists beyond the t Hooft limit.
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