We investigate percolation on growing networks where the evolution of connected components resembles a non-equilibrium version of the multiplicative coalescent. The supercritical $π> π_c$ regime for a host of such models was conjectured in statistical physics, and then rigorously proven in mathematics, to exhibit behavior similar to the BKT infinite-order phase transition as $π\searrow π_c$. It has further been conjectured that the entire regime $π<π_c$ for such growing networks are ''critical'' with power-law cluster size distributions having a non-universal exponent for all values of $π\in (0, π_c)$. In this paper, we study percolation on the uniform attachment model, as a concrete template in order to develop general tools based on stochastic approximation, local convergence, branching random walks and tree-graph inequalities to prove the above conjectured phenomena. For each $π\in (0,π_c)$, we show there exists an explicit $α(π) \in (0,\tfrac{1}{2}) $ such that the maximal component size, as well as the size of the component containing any fixed vertex, all re-scaled by $n^{α(π)}$, converge almost surely to strictly positive random variables as the network size $n \to \infty$. These dynamics lead to novel phenomena, compared to classical 'static' models, including long-range dependence and fixation of the identity of the maximal component, within finite time, among a finite number of 'early' components. Moreover, in contrast with most static network models, we show that the susceptibility, that is, the expected size of the component of a uniformly chosen vertex, remains bounded as the network grows and $π$ approaches $π_c$ from below. The general tools developed in this paper will be used in follow-up work to understand percolation for general growing network evolution models.