The most consecutive birdies I've had was 3. Achieved once at Overstone Park GC (3rd, 4th, 5th holes), and once at Leighton Buzzard GC (12th, 13th, 14th holes).
But the most memorable scoring stretch of golf holes I've played was -4 for 'the loop' on the Old Course at St Andrews, from the White tees in December 2016 :
Eagle on the par 4 7th (holed out from 120 yards)
Par on the par 3 8th (lipped out 12 foot birdie putt)
Birdie on the par 4 9th
Birdie on the par 4 10th
Par on the par 3 11th
Three in a row last year when my club was playing against Ratho Park at their course. I was playing in a fourball against their Captain and birdied the first three holes, the third being a thirty footer with an enormous bend that if I'd missed would have gone miles past. Walking down the fourth fairway the Captains partner said that if I birdied the fourth, he was walking off! I think he was joking.
Three in a row. Matchplay, and almost felt sorry for opponent as putts were going in from every where. Think it was six in the round, but it was about 40 years ago. These days very rare if I have more than one in a round.
Once played High Post as a visitor on my own, 2 members walked up on the 1st tee and asked if I wanted to join them. Started birdie, birdie, birdie......look on their faces was priceless
Probably no more than two successive birdies at my current course but I went birdie, birdie, eagle at Wimbledon Common, and then melted like an ice cream in the summer sun. Can recall topping a tee shot on the 7th (simple shot over a ravine and only about 150 yards carry) and losing a ball in the trees right on the ninth. Even back then, no consistency