It's briefly 8 at Wembley Park where the met lines become fast and slow and jubilee and Chiltern. Only for a short period is it eight tracks though.Six together between Bushey and on the approach to Euston and Finchley Rd and Harrow-on-the-Hill (albeit 4 LUL on the latter) are the first that come to mind - where they cross near Kenton is the only pass in Britain where two stretches of 6-track line do so.
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Waterloo-Clapham Jn as well.
There are loads around stations only that are 6 track or more too.
Redhill has 14 tracks side by side, if you include sidings which are completely overgrown and rarely, if ever, used (and which will probably, quite sensibly, soon be demolished).How about next to the old Willesden Freightliner terminal.
Up and Dn Relief (LLG)
Up and Dn DC
Up and Dn City
Quite a good total for six miles out from a major station.