The start station cannot be on the District or the Metropolitan.
Proof: you can't change from one to the other in one go, so they're both out as starting possibilities.
You need to start by narrowing down the lines which intersect with ALL the others. If you ignore the W&C (as all two of its stations are on other lines), the only lines which you can start on are the Hammersmith and City and Jubilee lines (any line not connecting with the ELL is out until the ELL is removed from the system and we know the District cannot be a starting line from above).
If you're going to say that you must be able to get to any station on any line at ANY time of the day, the challenge is impossible, as Chesham is only reachable from the H&C with one change during evening peak. Similarly, Olympia is something else that can't be done in one change as you need to change at HSK or Earl's Court and neither are on the H&C or Jubilee.
So, if we ignore Chesham (or allow someone to do it in the evening peak) and Olympia (or have it as the only exception to the rule), the answer is "any station on the H&C or Jubilee lines, so long as it's not an interchange station that uses the line you wish to travel to (or you won't actually change trains) AND so long as you start off on a H&C or Jubilee train from that station". Phew.
So, more specifically, any of the following: Goldhawk Road, Shepherd's Bush, Latimer Road, Labroke Grove, Westbourne Park, Royal Oak, Paddington (H&C), Great Portland Street, Euston Square, Farringdon, Barbican, Stepney Green, Bow Road, Bromley-by-Bow, Plaistow, Upton Park, East Ham, Barking, Stanmore, Canons Park, Queensbury, Kingsbury, Neasden, Dollis Hill, Willesden Green, Kilburn, West Hampstead, Swiss Cottage, St John's Wood, Southwark, Bermondsey, Canary Wharf, North Greenwich or Canning Town.
Try to go to anywhere from any of those stations starting on the H&C or Jubilee, and you only need ONE change to get to any other station (excluding the Chesham shuttle).
Does that work?
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