Keeping with my regional chauvinist theme: giving Merseyside as many League clubs as Greater Manchester (GM currently has seven).
There's the three current league clubs to start - Liverpool, Everton, Tranmere (the latter only just returned after a three-year absence, albeit with ninety-odd years of continuous membership beforehand).
Then there's the two former league clubs, Southport and New Brighton. The Sandgrounders were a League team until 1978 - and were actually the last team voted out of the Football League - and are currently bouncing around between the upper and lower divisions of the National League, but New Brighton were voted out in 1953 and have since ceased to exist entirely on two separate occasions. Given they were already the second iteration of League football in the town, New Brighton does not seem to like football.
After that it gets a bit tricky. South Liverpool (who were actually a phoenix club, the original club having become the second New Brighton team) applied for league membership ten times but never got more than a handful of votes. They were also exceedingly illustrious for a non-league side, winning the Welsh Cup in 1939, hosting the first match under permanent floodlights, and being the scene of
a charity fundraiser that pitted a side captained by Billy Liddell against one led by Ferenc Puskas that is the first place I am headed if they ever invent time travel. But again, the limited company responsible folded in 1991 and while the team haven't disappeared as completely as New Brighton they are currently languishing somewhere very far from the league pyramid.
That's six candidates so far; you'd have to dig pretty deep for a seventh. Runcorn FC would probably be the best bet, having won what is now the National League in 1982 - but they were barred from League eligibility to not meeting stadium requirements (given said stadium literally fell apart a few years later, this is not an unfair call), and once again they ceased to exist in 2006. Also, they aren't actually a Merseyside team. Prescot Cables and Marine both have lengthy and notable non-league histories but have never really showed any prospect of getting beyond the Northern League. Any further and we won't just be scraping the bottom of the barrel, we'll be falling in and drowning in the runoff.
In short: it might be doable, but it wouldn't be easy.