A question about gravity Some time ago I encouraged by 11 year old son to watch Cosmos: A Spacetime Odyssey with Neil DeGrasse Tyson, and subsequently Cosmos: A personal Voyage with Carl Sagan, as well as other astronomy shows including History Channel's The Universe.
One of these such shows inspired a question about black holes and gravity.
Can a black hole be so large that not even the effects of its own gravity can escape?
Perhaps the question rephrased might be: Why are gravitational waves immune to the effects of gravity?
 A: Gravitational* waves (GW) are not immune to the effects of general relativity (i.e. gravity).  If there was a source of GWs inside the event horizon, they would not emanate out of it (e.g. this answer --- which, I think, is much better than the accepted answer on that question).
At the same time, the 'information' about the gravitational field which is experienced/observed outside of an event-horizon has always been outside the event horizon --- and thus does not need to be 'communicated' out.
Consider, for example, if our sun collapsed entirely into a black-hole (e.g. this question).  The gravity outside of it wouldn't actually change (assuming the mass of the sun was completely preserved into the black-hole) at all.  The 'information' about the black-hole's mass is already (and has always been) outside of the event horizon.  If additional matter falls into the black-hole (e.g. this question), the 'information' about it still stays on the outside --- in particular, one can think of it as being imprinted onto the event-horizon.
Regarding your initial question of,  

Can a black hole be so large that not even the effects of its own gravity can escape?  

The answer is thus, no; but I'd also like to point out that black-holes are 'scale-invariant' --- in that none of the physics changes as the total size/mass increases.
Side-Note:
*'Gravity Waves' are (unfortunately) a term often used for certain types of sound-waves caused by gravity in stars, so the term 'gravitational waves' is preferable.  This usage is not universal.
