Ramanujan presented many identities, Hardy chose one which for him represented the best of Ramanujan. There are many proofs for this identity.
(for example, H. H. Chan’s proof, M. Hirschhorn's proof...)
Is there an elementary proof for Ramanujan's "most beautiful" identity?
$$\displaystyle{\sum_{n=0}^\infty p(5n+4)q^n}=5\frac{(q^5;q^5)^5_\infty}{(q;q)^6_\infty}$$ for $|q|<1$, where $p(n)$ is the partition function, and $(a; q)_\infty := \prod_{n \geq 0} (1 - a q^n)$.
Oops, I see that the OP actually refers to Hirschhorn's proof; it's not "elementary" enough...? Hirschhorn calls it a "simple" proof, and it indeed does not seem to require advance math.
Ramanujan’s “Most Beautiful Identity” by Michael Hirschhorn (2011) [download link]
Of all the 4000 or so identities Ramanujan presented, Hardy chose one which for him represented the best of Ramanujan. We give a simple proof of the identity. On the way, we give a new proof of an important identity that Ramanujan stated but did not prove.
