Abstract: Recent work shows that ordinary Reed--Solomon codes over prime fields admit polynomial-size lists up to list-decoding capacity on every evaluation set. We show a contrasting phenomenon over binary extension fields. For every fixed rate $0<R<1$, there exists $\varepsilon_R>0$ such that, for all sufficiently large $h$, the full-field Reed--Solomon code over $\mathbb{F}_{2^h}$ has a received word with $n^{Ω_R(\log n)}$ codewords within relative distance $1-R-\varepsilon_R$. To our knowledge, these are the first superpolynomial ordinary Reed--Solomon list-size lower bounds at fixed positive rate and a fixed positive agreement gap above the rate. Thus capacity-approaching polynomial list size for Reed--Solomon codes is not uniform over the underlying field, and the obstruction occurs at every fixed rate. As a puncturing consequence, the same holds for primitive Reed--Solomon codes; the obstruction is also robust under controlled dense puncturing. The proof begins with an exact correspondence that turns certain binary Bose--Chaudhuri--Hocquenghem weight-enumerator coefficients into sizes of single Reed--Solomon lists. A classical binary weight family gives the initial quarter-rate construction, while a quadratic-form enumeration supplies its superpolynomial multiplicity. A common-kernel quotient/descent argument and common-root padding then extend the construction to all rates. We also determine the optimized agreement envelope achieved by the construction.