I've had several sequences each lasting several days travelling without getting a real bed.

I've also had 6 international flights departing on the same day (AKL-SIN-CGK-SIN-CGK-SIN-NRT, yes 2 round trips to CGK with activities in SIN in between each arrival and departure).

Back to the original question, and all not counting repeats. 5 seems to be my limit.

3 in a calendar day too many times to count

4 within 24 hours - Italy (x2), Switzerland (x2), France, Vatican City

4 in a calendar day, a couple of times while bussing in Europe

4 in a calendar day flying - UK, Netherlands, Germany, Denmark
4 in a calendar day flying - Oman, UAE, UK, Spain
4 in a calendar day flying - NZ, Tonga, Samoa, USA
4 in a calendar day flying - Germany, Switzerland (x3), Luxembourg, Italy
4 in a calendar day flying - China (HKG) (x2), South Korea, Canada, USA
4 in a calendar day flying - NZ, Singapore, UK, Switzerland
4 in a calendar day flying - Germany (x2), Serbia, Swizterland, Tunisia
4 in a calendar day flying - NZ, Australia, Singapore, Maldives
4 in a calendar day flying - Brazil, Argentina, Chile, Ecuador

5 in a calendar day flying - USA, Barbados (x2), St Vincent, Trinidad & Tobago, Grenada
5 in a calendar day flying - Iran, Austria, Bosnia & Herzegovina, Croatia, Germany
5 in a calendar day flying - Algeria, Germany (x2), Slovenia, Kosovo, Austria
5 in a calendar day flying - NZ (x2), Australia, Chile, Paraguay, Argentina