Is there a limit on how much a lender can require in escrow?

Yes, when the loan is a federally related mortgage under RESPA. The CFPB says those limits apply both to the amount collected at closing and to the escrow portion of later payments. An escrow account is not required on every loan, but it is a way to set aside taxes and insurance so those bills can be paid when due.

Before closing, the lender estimates the annual expenses that will be paid from the account. It can collect enough in advance so the balance is not expected to turn negative, plus an extra two months of those estimated disbursements as a cushion. Each month after that, the servicer can require up to one-twelfth of the annual escrow amounts it reasonably expects to pay. Most loans with escrow also come with an initial statement and an annual statement showing history and the next year's projection.

A loan that is higher-priced under the Truth in Lending Act may require escrow for at least the first five years. Some loan types keep escrow for the full term. The lender or servicer is supposed to tell you if that applies. This answer restates the CFPB's published formula and is not a fixed dollar cap.