James Milner
When you talk about squeezing every last drop of production out of a player, Milner immediately comes to mind. Milner arrived at Liverpool at the age of 29, what some consider to be the beginning of the end of one's prime, but he contributed well up until the end, when he left at the age of 36.
In a way, there was no money lost on Milner, as he arrived on a free transfer as well. His value at that time was €14 million, and it only rose one million higher, a fittingly-boring arc for "boring James."
When he left the club, his value was only a million or so, but was still making appearances in all competitions at all positions, well worth the small fee Liverpool missed out on by hanging on through the end of his pact.
Joël Matip
Matip is the latest free transfer departure, as he's leaving the club this summer, although we don't yet know where. The Cameroonian international arrived on a free, and leaves the same way seven years later after a really solid Liverpool career. But could the Reds have cashed in a bit more?
Probably yes, but not by much. As was the case for so many players, his value peaked in late 2019, but there was no realistic sale opportunity then.
In the years since, however, plenty of Liverpool centre-backs have emerged alongside Virgil van Dijk, and Matip probably could have been moved for somewhere between €25 to €35 million in the year-plus after Liverpool won the Premier League in 2020.
Thiago Alcântara
Liverpool got a real bargain on Thiago- as per Transfermarkt, he was bought at a fee of €22 million despite a market value of €48 million. However, he's somehow failed to live up to even that modest price tag.
Technically, any moment would have been an optimal one to sell Thiago, as the club really never got much from him on the pitch, and his transfer value plummeted his entire time with the club.
You could say that this was a bad decision not to sell, but there's no one moment where it would have made the most sense- Thiago was just a bad player to bring to the club.