I take it we're talking transport authorities (shire county / unitary / whatever PTEs are called this week) not boroughs / districts that don't have transport responsibilities within two-tier counties / London / PTE areas.
In which case Berkshire has potential (the county council was abolished about 20 years ago and the existing small unitary councils were made unitary councils without merging them as happened in some bits of England.)
Courtney 53 Bracknell - Wexham Park Hospital serves 4 (Bracknell Forest, Windsor & Maidenhead, Bucks, Slough (although it crosses boundaries 5 times as it goes - eastbound - from Bucks to Slough then Bucks to Slough again.)
Reading Buses' Green Line 702 is marketed as limited stop, but is registered local bus service throughout (although it may be registered in two chunks because of the 50 Km drivers hours rule - it was under First) - if you treat it as a single route, it manages 6 on the journeys that run to / from Reading (Reading, Wokingham, Bracknell Forest, Windsor & Maidenhead, Slough, London) - although it doesn't have any bus stops in Wokingham Borough (it runs along the A 329 M / A 329.)
If we're talking all local authorities, it must clock up a few London boroughs between the Slough / London boundary and Victoria.
Bee Line 192 when it ran Reading - Heathrow also managed 6 but this was split in to smaller chunks in the early 2000s.