Making use of the Heathrow Freezone, I think it can be done for £1.50, in theory at any rate. Take any bus to the edge of the Freezone at Harlington Corner or Hatton Cross. That costs nothing and you don't need to touch in.
Then, buses 81 or 222 from Harlington or 482 from Hatton Cross to Hounslow West, followed by the H91 to Turnham Green can, according to Journey Planner, be done in less than an hour (50 minutes from Hatton Cross, a bit more from Harlington). With a bit of luck a 27 will turn up at Turnham Green before your hour is up and then you're away - right through Zone 1 and out to Chalk Farm if you so desire.