Do you always start counting days the day after the triggering event?

For example:

A CTR has to be filed by the 15th day. So if the transactions occur on the 1st, Your last day to file is the 16th.
If you receive an application and deny on March 1st, the last day to notify the customer and meet the 30 day deadline is the 31st.
Mortgage application on the 1st, last day to deliver the loan estimate is the 4th (assuming all business days involved).

Are there any regulatory requirements that specify another way to count days until the deadline?

None that I am aware but every regulation could specify their own rules.

In the wonderful world of laws & regulations, it's very risky to think in the terms "always" and "never." As Randy says, each law, rule, and regulation stands alone--possibly containing its own definitions and specifications. You should "always" begin with the onboard definitions, if any. In Reg. Z, for example (Appendix J, paragraph b(5)(i)), a day is defined as "the number of 24-hour intervals between any point in time on the first date to the same point in time on the second date." I'm not aware of any other case where that definition is valid.

For many (most?) purposes, a "day" is the 24 hour period beginning at midnight. That's the case with Fed Funds transactions and most other loans and deposits with a running interest clock...and it's always been my fall-back rule of thumb for measuring the passage of time. Until midnight, it's "today." When Cinderella's coach turns back into a pumpkin, that's "tomorrow" and the day-count increments by one.

The exact filing requirement for CTRs is: "1010.306 Filing of reports, paragraph (a)(1), A report ... shall be filed by the financial institution within 15 days following the day on which the reportable transaction occurred."

Since this reporting deadline is tied to tran dates, then you begin with the tran date as day 0 and count forward in calendar days.

And now add to that state laws. I recall and issue I had with many lenders was that the first payment on an instalment had to be anywhere between 15 days and a month and 15 days. Many consider that 45 days, but not so. And if you stopped there you'd end up off my X-mas card list. TX law for this defines a month as from the closing date to the same date of the next month, plus up to 15 days. And if the next month didn't have that date, use the last day of the month. So Jan. 31st to Feb 28th is a month.

What about Reg E ? Do we start counting investigation days on the date consumer asserts the error? Or the day after? What should be day 1?

The date of notice is Day 0. Day 1 is the first business day after the date of initial notice.

Because a "business day" is defined in Reg E as a 24 hour period beginning at midnight, the day you get a claim is day zero, as Brian noted, unless that happened be the exact moment you got the claim. Completely unlikely but hey, we pick nits for a living.