In the following sentence, what is the meaning of whole?
I have a whole bunch of stuff to do this morning.
I understand the meaning of "a bunch of something," but I am not clear why whole should be used in that sentence.
This is a modern idiom (esp. US, informal). The following two sentences are roughly equivalent:
I have a whole bunch of stuff to do this morning.
I have lots of things to do this morning.
The idiom "whole bunch of stuff" (also bunch of stuff, whole ton of stuff etc) is a somewhat informal way of saying "lots of things".
It alludes to the large number of things [that need to be done this morning] without being specific as to what they are or specifically how many of those things there really are - for example because the speaker merely wants the listener to be aware that the speaker is busy, without necessarily wishing to brief the listener on the particularities of what the speaker is doing. This might be because:
The CEO can't see you today, Mr. Jones. He's got a whole bunch of stuff to do this morning and has asked not to be disturbed.
Dave - can you make sure that I'm not disturbed for the next hour or so? I've got a whole bunch of stuff I need to sort out before our meeting with the Senator.
We got to meet with the President, but he was a bit delayed because the CIA were briefing him on a whole bunch of stuff and their meeting overran.
I work much harder than Alice. I mean, this morning I did a whole ton of stuff whilst she was chatting with her boyfriend on the phone.
OP's whole bunch of stuff gives an impression of "uneducated" speech. A more "acceptable" informal version would be, for example whole lot of things (that's 319,000 written instances in Google Books).
In such constructions, whole is being used somewhat loosely as an intensifier (meaning approximately complete/entire, with connotations of large/substantial).
It is used for emphasis. Look at these dictionary examples. (They include "a whole bunch.")