How do I phrase an email requesting more money when I've already accepted a specific budget? The client is asking for more time than I would have expected to spend asking for revisions upon revisions and well past the initial deadline. It's a new client.
If you have yet to agree that extra work is billable then learn for your next contract. "One round of revisions and two weeks post delivery bug fixes for work agreed at the outset. All other out of scope changes shall be billed at my hourly rate of x ..."
You could try the tactic for now of I will take these revisions listed and deliver next Friday. After that any changes are new work. That way you are swallowing it allowing good grace but being firm. It's hard some times. Esp when clients start thinking of you as coworkers and just drift reply to you.