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.
*'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.