This is a bit complicated. I would start by summing up the events by day:
CREATE VIEW BView AS
SELECT UserID, BevID, CAST(EventDateTime AS DATE) AS EventDate, COUNT(*) AS NumEvents
FROM beverages_log
GROUP BY UserID, BevID, CAST(EventDateTime AS DATE)
Then I used the Dates table (only a table with one row per day, very convenient for) to examine all possible date ranges and drop any with a space. It will probably be slow as hell, but this is the beginning:
SELECT
UserID, BevID, MAX(StreakLength) AS StreakLength
FROM
(
SELECT
B1.UserID, B1.BevID, B1.EventDate AS StreakStart, DATEDIFF(DD, StartDate.Date, EndDate.Date) AS StreakLength
FROM
BView AS B1
INNER JOIN Dates AS StartDate ON B1.EventDate = StartDate.Date
INNER JOIN Dates AS EndDate ON EndDate.Date > StartDate.Date
WHERE
B1.NumEvents >= 5
AND NOT EXISTS (SELECT * FROM Dates AS MissedDay WHERE MissedDay.Date > StartDate.Date AND MissedDay.Date <= EndDate.Date AND NOT EXISTS (SELECT * FROM BView AS B2 WHERE B1.UserID = B2.UserID AND B1.BevID = B2.BevID AND MissedDay.Date = B2.EventDate))
AND NOT EXISTS (SELECT * FROM BView AS B2 WHERE B1.UserID = B2.UserID AND B1.BevID = B2.BevID AND B2.EventDate > StartDate.Date AND B2.EventDate <= EndDate.Date AND B2.NumEvents < 5)
) AS X
GROUP BY
UserID, BevID
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