We propose a model that generates a new class of networks exhibiting power-law degree distribution with a spectrum of exponents depending on the number of links ($m$) with which incoming nodes join the existing network. Unlike the Barabási-Albert (BA) model, each new node first picks an existing node at random, and connects not with this but with $m$ of its neighbors also picked at random. Counterintuitively enough, such a mediation-driven attachment rule results not only in preferential but super-preferential attachment, albeit in disguise. We show that for small $m$, the dynamics of our model is governed by winners take all phenomenon, and for higher $m$ it is governed by winners take some. Besides, we show that the mean of the inverse harmonic mean of degrees of the neighborhood of all existing nodes is a measure that can well qualify how straight the degree distribution is.