Some bodybuilders use aspirin because it lowers cortisol effects (not production, just effects). But it also knocks t and GH levels at the same time so the whole thing is kinda redundant. The effect on protein synthesis relates to this hormonal profile change too I believe. Unless you have just been chased by a dinosaur and you're worried about failing your upcoming cortisol level test aspirin supplementation is not for you
If you have to use aspirin or similar regularly for something just follow the instructions on the box. They're conservative for a reason. I take aspirin once a week on average when my patellar tendons flare up no worries. The levels used to elicit a hormonal response in studies are pretty big. If you're gobbling that much over the counter pain med regularly you best get some professional help.