...I could get a couple of 4s and 5s from the Underground map, but alas the East Coast Main line is the winner with 8.
Alnmouth
Berwick-upon-Tweed
Dunbar
Edinburgh Waverley
Haymarket
Inverkeithing
Kirkcaldy
Leuchars
But that's missing out loads of stations - Drem, Longniddry, Prestonpans, Wallyford, Musselburgh, South Gyle, Edinburgh Gateway,, Dalmeny, Noth Queensferry, Dalgety Bay, Aberdour, Burntisland, Kinghorn, Markinch, Ladybank, Springfield, and Cupar.
"Go through consecutively" sounds to me like those should be counted - if you can leave stations out to build a sequence it gets too easy, unless you impose something else - like these being stops on one service.