You cannot add a Google Contacts group to another group directly. Groups do not nest (kind of like how Gmail labels do not nest if you have not activated nesting in Labs). However, we don't need to add users to BIG one-by-one either. You have a couple options, actually.
The Easy Way
The most direct, and probably easiest for most circumstances, is to batch add users from A to BIG, then from B to BIG, then from C to BIG (etc.). I assume you already have A, B, and C, so now create BIG. Now go to A. Click the "Select All" checkbox to select every contact in Group A. Then, click the Groups menu (the drop-down menu with a three-person icon) and click the open checkbox next to the BIG group. This will add everyone from A to BIG.
Removing contacts from lists can be done on a one-by-one or group-by-group basis (or any combination thereof).
Something More Elaborate
UPDATE: Google no longer permits nested groups. No workaround at this time. My original notes still included here for reference purposes, and in the hopes that we find a workaround. Made my answer a wiki to facilitate other solutions!
"Hang on!" you say. (I don't know, what do you usually say? It's a one-sided conversation right now, I get to put words in your mouth. Here come some more:) "This easy way works for simple lists, but what if things get complicated? What if Jane Test is leaving Group B. Normally, this would mean that I remove her from BIG also. But I have lots of contacts and groups; Maybe Jane should still be in BIG because she is a member of Group C also. Or, what if I forget about this issue and don't check Jane's contact card for other groups when editing it, and then take her out of BIG mistakenly?"
Now I think you're being picky, but there is a solution for this too. Instead of using Google Contacts for these more complex contact/group and group/group relationships, create some private Google Groups for A, B, C, and BIG. Unlike Contacts groups, Groups groups (is that a thing?) can include other Groups groups. (See: http://support.google.com/a/bin/answer.py?hl=en&answer=167100) Look, we're just going to call those "Groups" now. You can email everyone in a Google Group by setting their member preferences (for that Group) to receive all posts by email, and then by emailing a post to the group. Every member of the group gets the email.
That way, if Jane is in Groups B and C; and Groups A, B, and C are "members" of Group BIG; then when you remove Jane from Group B she will retain her membership in Group BIG in virtue of her membership in Group C. So, you don't have to check and you are less likely to mistakenly remove Jane from BIG.