It's Certainly Possible
The Clinton campaign made big noises after she won Pennsylvania (This post is can be filed under the "better late than never" tag). They said she'd raised 10 million dollars in 24 hours. 100,000 contributors, 80,000 of them new, with an average donation of about $100.
Let's say you are really rich, and you want to donate millions of dollars to a candidate. FEC regulations prevent that. But you are really rich, isn't there a way? <looks around> sure there is, and I'd be surprised if all major campaigns don't do this.
They are called "unitemized deductions" and they are totally legal under FEC rules. For starters, to enforce the campaign finance limitations, campaigns have to report their donations. Um, except there is a huge loophole. If you donate less than $200 in a single campaign cycle, the campaigns don't have to report your name. So, in this age of computers, all a rich person needs to do is hand a million dollars to the candidate, and the candidate just reports to the FEC that it was all new donors and under $200 each, and so no further paperwork is required. Yes, a compliant person in the person's finance committee is required, but the Finance Chairs of campaigns are often hired for their connections, rather than their skills.
I suspect the DNC, RNC, and all major campaigns receive donations this way. Here is a snippet on the RNC.