In the `Number-on-Forehead' (NOF) model of multiparty communication, the input is a k \times m boolean matrix A (where k is the number of players) and Player i sees all bits except those in the i-th row, and the players communicate by broadcast in order to evaluate a specified function f at A. We discover new computational power when k exceeds \log m. We give a protocol with communication cost poly-logarithmic in m, for block composed functions with limited block size. These are functions of the form f \circ g where f is a symmetric b-variate function, and g is a k r-variate function and f \circ g(A) is defined, for a k \times br matrix to be f(g(A^1),\ldots,g(A^b)) where A^i is the i-th k\times r block of A. Our protocol works provided that k > 1+ \ln b + 2^r. Ada et.al (ICALP'12) previously obtained \emph{simultaneous} and deterministic efficient protocols for composed functions of block-width r=1. The new protocol is the first to work for block composed functions with r>1. Moreover, it is simultaneous, with vanishingly small error probability, if public coin randomness is allowed. The deterministic and zero-error version barely uses interaction.