To address the feature request itself: This feature should not be implemented.
Unlimited spam flags would be bad, because spam flags are extremely powerful.
- A spam flag carries an automatic downvote from the system. This doesn't decrease the author's reputation, but will alter other users' perceptions of the post, and these votes combine with actual downvotes to magnify the usual consequences of downvoting. For example, if a question is -2, and you spam flag it and downvote it, it is now at -4 and it does not appear on the main page (until it is upvoted or undownvoted, or the spam flag expires).
- If a post is flagged as spam three times, it cannot appear on the main page. (This remains in effect until the flags expire.) So three people (or two people or one person, taking sockpuppetry or account-hacking as possibilities) can cause severe harm by spam flagging a large number of posts.
Instead, the number of spam flags you have should, and does, increase with the amount of helpful flagging you've done.
When people don't have enough spam flags to handle all the spam that is coming in, the solution is to contact moderators through other means. As has been said, chat is one way. Moderators are often there. You can ping them. You can even ping moderators who are not there--they will be notified if they've been in the room anytime during the last couple of days. (They may take considerable time to respond to the notification, though.) If you can't find a moderator, you can still likely use chat to find other users with remaining spam flags. Thus, with or without a moderator present, chat can be used to organize spam fighting efforts.
If you can raise neither a moderator nor any user with remaining spam flags in chat, you could use a custom moderator attention flag on one spam post to ask that a moderator come to chat to help with a new wave of spam, and/or give a list of URL's pointing to the profiles of user accounts created to post spam (and optionally, links to some of their spam posts).
One possible alternative spam flagging feature that could be implemented without nearly as much danger as the feature requested here, would be to make spam flags work more like comment flags in the way the number of flags you have at any given moment is determined. When a spam flag is marked as helpful by a moderator, either manually or by deleting the spam-flagged post (but not under any other circumstance, to avoid giving additional spam flags to people who are in the process of abusing them in ways detailed above), the spam flag could be returned to the user. That is, each spam flag marked helpful by a moderator could be used again that same day. Then, users would be able to flag more spam per day.
Of course, that relies on moderators being available at least some of the time. But it would help at least some of the time, even with spam that appears when no moderator is readily available.