What could be the problem with my 2005 Honda Accord, I have a starting problem?

I recently brought a 2005 Honda Accord.  
After leaving the sales place I stopped to
the service station to refuel. I turn off the
engine,  try to turn the car on didn't start.  I
purchase a new battery,  car started.  Two
weeks later I started to experience starting
problems having to play with my terminals
to start the car. However, that didn't work
anymore so I had a local mechanic check it
and change the starter,  car ran find that
day. Next morning I experience the same
problem car won't start unless I played with
the terminal.  Took the car back to the
mechanic they check battery, starter,
alternator and wires everything was good.
Later that evening car didn't start not even
when I played with the terminal. What could
be the problem?