As a medical technologist, I can tell you that in Quebec at least, you are absolutely not allowed to order tests on yourself. Only a doctor can fill out test requisitions, and the patient, nurses, or technologists aren’t allowed to just add other tests on there because they want to.
We get a lot of requisitions in the hospital test center, where we draw blood on outpatients, with tests checked off in a different color pen, or scribbled in by hand, and we’re supposed to contact the doctor in question and ask about it. In reality, we don’t always have time to do that, but extra, unnecessary tests are an added cost to the health care system.
I have no clue about the legality/feasibility of this sort of thing in a private healthcare setting, however. Seems to me like it’s a bad idea, because then if you’d be getting results, without a doctor discussing them with you, you’re more likely to misunderstand what the results mean, and possibly overreact if something seems out of range. Sometimes “abnormal” doesn’t necessarily mean there’s a problem - but only your doctor would know that, because he’s seen you and evaluated your condition based on many factors outside of those tests.