Ashford (Surrey) to Ashford (Kent) - clearly no direct train in passenger service, but I would like to challenge the idea that there is 'no connecting track' - i.e. that the journey could not be made by rail. A potential route is
Ashford (Surrey), London Waterloo (reverse - I believe the reversal could not be done at Clapham Junction due a lack of line connection there), Havant (via Guildford - reverse at Havant, if for any reason not feasible, continue to reverse at Portsmouth and Southsea), then West Coastway followed by East Coastway, to end up at Ashford (Kent), i.e. Ashford International.
Another option would be
Ashford (Surrey), then via Chertsey to Woking and as above.
Or
Ashford (Surrey), Reading, Basingstoke (subject to track connection at preceding signals to go into something other than the usual bay platform, I believe it's OK as XC services do it?), then via Eastleigh, Hedge End as as above.