Well, how about this? I liked the Biggin Hill bit, but if you're going to be like that, here's one valid under the challenge rules, with even more means of transport...
Heathrow Terminal 5 Business Car Park

Heathrow Terminal 5 (Heathrow Pod)

Heathrow Terminals 123 (via Pic)

Paddington (Heathrow Express)

Ealing Broadway (Heathrow Connect)

Greenford (FGW)

South Ruislip (Cen)

Marylebone (Chiltern)

Edgware Road (foot)

Baker Street (Cir)

Great Portland Street (Met)

Regents Park

Waterloo (Bak)

Bank (W&C)

Euston (Nor)

Birmingham New Street (London Midland)

Edinburgh Waverley (Virgin)

King's Cross (East Coast)

St Pancras International (foot)

Luton Airport Parkway (East Midlands Trains)

Luton Airport (bus)

Dublin Airport (Ryanair)

London Southend Airport (Aer Lingus)

Southend Victoria (Greater Anglia)

Southend Central (foot)

Fenchurch Street (c2c)

Tower Hill (foot)

Aldgate East (Dis)

Whitechapel (H&C)

Canada Water (LO)

North Greenwich (Jub)

Royal Victoria (Emirates Air Line)

East India (DLR)

Blackwall Tunnel Approach Bus Stop (foot)

Lewisham (108 bus)

Elmers End (Southeastern)

East Croydon (Tramlink)

London Bridge (FCC)

Clapham Junction (Southern)

Vauxhall (SWT)

Victoria (Vic)

Gatwick Airport (Gatwick Express)

Dubai International Airport (Emirates Airline)

Amsterdam Airport Schiphol (KLM)

London City Airport (BA Cityflyer)