My completely off-the-wall idea is to join up Crossrail with the
Hammersmith & City and run Crossrail trains (from Shenfield) that
would otherwise terminate at Paddington on to Hammersmith. After all,
Crossrail as proposed is just another tube line - Shenfield is closer
than Amersham, hardly further than Epping or Upminster.

The pattern could be:

4 fast Heathrow
4 slow Heathrow
8 Maidenhead / Reading
8 Hammersmith

Or 12 Hammersmith if Ken can't get BAA to agree to HeX on Crossrail.

For Crossrail it would save all the expense of the sidings and ghost
station needed to satisfy Health & Safety re passengers staying on
terminating trains at Paddington.

There would be a huge benefit to the Circle Line. All Circle and
Wimbleware trains would continue beyond Edgware Road, providing for
the first time since 1868 a proper much-needed frequent service from
west to north Circle, dramatically improving performance by removing
the conflicts and bottleneck west of Edgware Road. Running as a
Teapot line (Wimbledon - Edgware Road - Aldgate - High St Ken -
Edgware Road - Aldgate and vice versa) would solve the Circle problem
as well.

The two lines virtually intersect at Paddington anyway so joining them
shouldn't be too difficult. Platform lengthening and ancillary works
at the H&C stations, plus a new Royal Oak, would be challenging, but
no doubt not insurmountable.