I ran a couple of colonies (in poly hives) on double brood boxes last year without a QX. The mistake I made was using full size frames in the brood chambers and shallow supers above. In the height of the OSR season the queen laid up the lower super as well as the double brood boxes. When one colony decided to try swarming I was then stuck with the shallow frames of brood looking for a home.
This spring I'm going to repeat the experiment but using medium (6.25") LS frames throughout the hive. I've got one on these frames already and the others I'll do a shook swarm on to transfer them. I can't see them laying up more than 3 boxes and then only for a short period. The rest of the year they should be on two as two medium LS frames are larger than a 14*12, Jumbo LS or Dadant. I am also going to use one piece plastic frames which give more brood area compared to wooden frames. Each one is larger than a BS National full depth frame in area (101 sq ins).
Having the same size frames should make it easier to split and distribute the frames if I need to do an AS. The alternative would be using full size frames throughout (as I understand they do in Denmark when operating without a QX) but my extractor won't take them radially and of course they are very heavy.