There's generally no problem running a few preamp tubes pulling 1 mil each off a doubler. Usually there is no problem running an output section on a doubler, either.
But you may be disappointed with the performance of a tripler, meaning, unhappy with the behavior under load. Nevertheless, it's IMO a worthwhile learning experience to breadboard such a thing (leave your leads long!) to see if you can make it work. I'd urge you to breadboard such a thing (a tripler) and get some fat power resistors to simulate loading & see how it reacts to heavy load demand before committing to the design and thus the tranny.
I am not saying it's impossible. From my experience, a few things:
I call the caps inside the rectifier section "separator" caps. They maintain separation of the phases of the input signal and stack them up on top of each other. Because they are typically in series, their ufds get cut in half. Thus their charge-storage and smoothing capabilities are cut in half. Sometimes this cannot be overcome, even with fat caps on the rectifier output. I have found that one wants to place extra caps at the output of the rectifier section; and these, apart from increasing your parts costs, also start to take up real estate if your chassis is tight. Indeed, the caps inside the tripler are sometimes drawn on, current-wise, so much that even overly fat caps on the output can't overcome a bad ripple reaction under load. (though, maybe you like "sag")