A common grammar lesson that was taught to me in the US and that I've had to teach abroad in EFL classrooms is that we're not to use adverbs of emphasis with absolute modifiers, just as we're not supposed to use them as comparatives or superlatives.
Classic egregious examples of this mistake include very unique and more perfect, which seem obviously flawed to me. Other instances of further modified absolute modifiers are similarly meaningless or contradictory in nature.
I wonder, though, if the rule that I learned isn't overly broad, and whether it's taught similarly in the UK and elsewhere. I find myself modifying absolute modifiers quite regularly, and in some circumstances I think it's not only logical and correct, but quite meaningful. Phrases like almost exactly or virtually all or practically infinite possess a specific meaning that can't really be replicated using nonabsolutes like very nearly or most or quite long.
In my own speech and writing, I use such phrases exactly as I see fit, so this isn't a request for permission, per se. I'm more curious about whether there's a more appropriate rule regarding absolutes that you rely on (and might be used in a classroom).
Best Answer
What you are doing in the examples very unique, more perfect is using an adverb to intensify a strong adjective; it seems a fairly good rule of thumb that this is often stylistically inadvisable. There is a certain inflation of the intensity of the adjectives, and the adverbs can be accused of pleonasm (as in "a triangular triangle"). This rule is often referred to on both sides of the Atlantic, and in many other languages too.
However, your other examples do more or less the opposite: you use adverbs to weaken (or qualify) the adjective in almost exactly, virtually all, and practically infinite. That is quite different: there is no inflation of the intensity of the adjectives themselves, nor are the adverbs pleonastic.
Other concerns may arise, though, in the case of absolutes and classifications (I like Fumblefingers' categories). Sometimes it is not very satisfying to qualify the unqualifiable, as in she's a bit pregnant: she's either pregnant or she isn't, so it is probably better to say she looks a bit pregnant, if that's what you mean. I think that's almost exactly the same thing, while perhaps acceptable, is usually not to be preferred over that's almost the same thing: the adjective feels a bit wordy and is made redundant by the adverb. The examples with all and infinite are what Fumblefingers calls absolutes, which I think should be qualified without qualms, where appropriate.