$p$-adic valuation of sums of distinct factorials: bound $f(a,p)$ or force it to infinity (Erdős #404)

Fix an integer $a\ge 1$ and a prime $p$. Over all finite strictly increasing sequences $a=a_1<a_2<\cdots<a_n$ of integers, consider the largest $k$ for which $$p^k \mid (a_1!+a_2!+\cdots+a_n!).$$ Question (i): for which pairs $(a,p)$ is this $k$ bounded (a finite supremum over all such sequences)? When it is bounded, let $f(a,p)$ denote the greatest attainable $k$, and describe the behaviour of the function $f(a,p)$. Question (ii): is there a prime $p$ and an infinite sequence $a_1<a_2<\cdots$ such that, writing $p^{m_k}$ for the exact power of $p$ dividing $\sum_{i\le k} a_i!$, one has $m_k\to\infty$?

Acceptance. FULLY RESOLVES: a complete proof (machine-checkable preferred) that (i) characterises exactly which pairs $(a,p)$ yield a bounded $k$ and, where bounded, determines or sharply bounds $f(a,p)$, and (ii) either exhibits a prime $p$ and an explicit infinite sequence with a certificate that $m_k\to\infty$, or proves that no such prime/sequence exists. ADVANCES (each independently checkable): (a) determine $f(a,p)$ exactly for a specific pair whose value is not currently known, or improve/complement Lin's bound $f(2,2)\le 254$ (e.g. prove it sharp or lower the ceiling), with a reproducible exhaustive-search certificate; (b) a new record lower bound on some $f(a,p)$, given as an explicit sequence together with its verified $p$-adic valuation; (c) a rigorous proof that $k$ is bounded (or that it is unbounded) for a specified infinite family of pairs $(a,p)$. State any numerical bound strictly against the best bound in the background. Deliver the explicit sequence plus valuation certificate, the proof file, or the search code plus attained bound.