We consider the fair division of indivisible items among $n$ agents with additive non-negative normalized valuations, with the goal of obtaining high value guarantees, that is, close to the proportional share for each agent. We prove that partitions where \emph{every} part yields high value for each agent are asymptotically limited by a discrepancy barrier of $Θ(\sqrt{n})$. Guided by this, our main objective is to overcome this barrier and achieve stronger individual guarantees for each agent in polynomial time. Towards this, we are able to exhibit an exponential improvement over the discrepancy barrier. In particular, we can create partitions on-the-go such that when agents arrive sequentially (representing a previously-agreed priority order) and pick a part autonomously and rationally (i.e., one of highest value), then each is guaranteed a part of value at least $\mathsf{PROP} - \mathcal{O}{(\log n)}$. Moreover, we show even better guarantees for three restricted valuation classes such as those defined by: a common ordering on items, a bound on the multiplicity of values, and a hypergraph with a bound on the \emph{influence} of any agent. Specifically, we study instances where: (1) the agents are ``close'' to unanimity in their relative valuation of the items -- a generalization of the ordered additive setting; (2) the valuation functions do not assign the same positive value to more than $t$ items; and (3) the valuation functions respect a hypergraph, a setting introduced by Christodoulou et al. [EC'23], where agents are vertices and items are hyperedges. While the sizes of the hyperedges and neighborhoods can be arbitrary, the influence of any agent $a$, defined as the number of its neighbors who value at least one item positively that $a$ also values positively, is bounded.