We obtain the first term in the high signal-to-noise ratio (SNR) asymptotic expansion of the sum-rate capacity of noncoherent fading networks, i.e., networks where the transmitters and receivers—while fully cognizant of the fading law—have no access to the fading realization. This term is an integer multiple of log log SNR with the coefficient having a simple combinatorial characterization. It can be interpreted as the effective number of parallel channels that can be supported by the network, i.e., as the maximal number of point-to-point single-user scalar channels that can be supported by the network in a manner that will allow, with proper power allocation, negligible cross interference. The results hold irrespective of whether the transmitters can cooperate or must operate in an multiple-access regime; irrespective of whether feedback from the receivers to the transmitters is available or not; and irrespective of whether the receivers can cooperate or not.