Hmmm, as a german business jet pilot that visits LCY regularly, I´ve got to say I find it hard NOT to query the clearance to 4000ft - when you go upstairs to pay your fees and/or use the crewlounge, there is plenty of "avoid levelbusts" / "remember the stepped clib" / etc posters on the walls. Also LCY JetCentre sends out good info on the subject to all users.
Sometimes the handover to Thames is quite late and so the "gate" of the initial call with cleared level might not work always. If we go outta there, it usually comes together with level off and first turn...
Someone asked why some of us climb so hard when in LCY...most likely because we have been told of numerous noise monitoring stations. Thats what I try to avoid, getting a prob with these things (and adhere to climb gradient requirements of course)