OP is asking if a case will stretch to the maximum headspace dimension after a single firing or does it require several firings to attain that condition.
It is certainly a function of the case construction and the hardness of the material.
Yes and more. Firstly how much 'slack' is there in the new case to chamber fit; secondly the powder charge's fill-ratio and its burning characteristics. In most situations, I'd reckon one or two firings, but it can be more. I realised this recently with a new 6.5mm Grendel Mini Howa and new Lapua brass. My first couple of trial load work-ups used two very different recommended powders under the same bullet - IMR-8208XBR with 100% fill-ratio and at top load slight compression; Ramshot Tac ball type powder with a top charge that only just broke 90% fill. MVs weren't that far apart and using QuickLOAD, neither set should generate any problem pressure levels and should also have been producing reasonably close ranges.
Extraction and bolt opening and case appearance were very different though. The Tac loaded cases had very easy primary and full extraction as expected, but primers were pancake flat for all charge weights. The XBR loaded cases had hard primary extraction for all charge weights, but rounded primers suggesting low to modest pressures (allied to the modest MVs and lowish QuickLOAD forecasts).
Head scratching time! Flat primers plus low to modest pressures usually = excessive headspace. Conversely, the XBR loaded brass showed all the signs of cases that were a longitudinal crush fit in the chamber - very distinct circular marks on case-heads imparted by the bolt turning on a case hard up against it until primary extraction was complete. There were no pressure signs on the case such as excessive expansion in the web area, and as already said very round primers with nice deep and round firing pin strike indentations. So, maybe the two loads had fireformed the brass differently? Decap some cases and get the calipers, comparator and appropriate Hornady 'headspace gauge' out. (A misnomer as it's a base to case shoulder comparator tool, nothing to do with true headspace gauges.) The XBR case readings were 0.005" longer than for the Tac cases. (Eh?

!! ) Scratch head a bit more! Measure new cases out of the box - they're a full 0.008-0.010" shorter than the fired XBR readings and around 0.004-0.005" shorter than the Tac cases.
So, we have new brass with a very short base to shoulder length compared to the chamber on a new rifle! (I assume this is deliberate on Lapua's part due to the cartridge's primary use being in semi-auto ARs; maybe a mismatch between US SAAMI and European CIP specs, but that seems very unlikely.) In any event, it means a lot of shoulder movement on the initial firing or firings. The two powder types and their fill-ratios apparently then acted differently in this situation despite MVs and calculated pressures being pretty similar, one (Tac) only doing half the fireform job; the other (XBR) not only doing it fully but ramming the shoulder forward so far and hard, the case remained a tight fit in the chamber.