"There is no evidence to support the benefit of routine early treatment with thyroxine in non-pregnant patients with a serum TSH above the reference range but <10mU/L. Physicians may wish to consider the suitability of a therapeutic trial of thyroxine on an individual patient basis" [1]
So what might happen to you if your TSH is, say 3.5, for at least a decade (hey hey, I know the answer to this one...)? Well, you're at risk of elevated cortisol with resulting mental illness, to which my previously posted links will attest. Doctors, if they're not especially attentive, will just prescribe the usual anti-depressants (perhaps a low dosage as an anxiolytic) which are band-aids and don't get to the root of the problem.
Note that I never mentioned non-standard tests - "keep testing" means "keep getting the standard tests". The problem is that interpretation of the standard tests varies - some test results may or may not indicate disease, and this is not always carefully followed up on. There are of course many more people in this grey area than there are people with overt disease. I'd say if you're in the grey area and you have chronic mental health symptoms the risk/reward ratio is overwhelmingly in favour of treatment with thyroid hormone (carefully monitored of course).
[1] http://www.british-thyroid-association.org/info-for-patients...