Climbing sling(s) – those are multi-purpose & inexpensive. For example, you can put them over a branch for pull-ups, or you can hold on to them for stabilisation when doing pistols, or you can use them for hamstring stretches
TRX Suspension Trainer – Not cheap, but pretty versatile and mobile training tool. Make sure to get it together with the door anchor. If you care less about mobility, then rings might be a better (and possibly cheaper) alternative
Kettlebell – Probably the best value-for-money (and space efficient) weight training tool. Ideally try them out in your gym, and go for one size bigger than you are comfortable with from the beginning so that it last you for a while (so if you start with a 16kg/1pood kettlebell as most men should, buy a 20kg one for home use)
Fingerless gloves – I know, I know, they are for sissies, but still. Sometime your hands just hurt.
Climbing rope – I always have some rope with me when exercising, especially when going to the park. The thin one (1-2mm, at least 100kg) does not take much space and can be used to fix the elastic bands. The thicker one (5mm+) can be used to fix your TRX on a high tree or the like, and you can use it for tightrope walking and climbing. I also always have a short (1-2m) piece of it with me which often helps fixing things (TRX on a thick tree, weights on my waist)
Carabiner – Particularly useful to fix the elastic band – using a cord (especially a thin one) directly might wear it down too quickly