(Native American English speaker here.)
Below is my thought process as I read the passage. I did not read ahead before writing each bit of explanation below: each description is what I thought, only having read what's quoted so far. This is interpretation by a lay reader; I have no special knowledge of or experience with legal writing.
By the way, as you will see, there are even more than three negations here.
However, it is possible for Acts to create offences
What's coming is an exception to something that came earlier, which probably said that Acts can't create a certain kind of offence.
that do not require any mens rea, or guilty mind, in order for the defendant to be convicted.
OK, now I see. The previous context suggested or implied that Acts (by Parliament, I assume) can't create offences that are defined solely by what someone did, without regard to intent. For example, a person might trespass without intending to, and an Act could make this an offence with a prescribed punishment. (This helps a lot as you're reading abstract material: pause and think of at least one example.)
The word "that" restricts the scope of "Acts".
But it is right for society to be wary about the number of such offences,
"But" indicates an exception to what came before. The word "wary" indicates that this exception is not a rule but a guideline suggesting that defining "such offences" contains a lot of potential for abuse. Society should watch out for a danger that is about to be described. If the law defined too many of these offences, something bad would happen.
so the courts presume that a statute does not impose criminal liability
Presumption means what you assume before you have more information or evidence. Right now, I'm a little confused. "So…" is probably short for "so that the…" But I can't tell yet if we are describing: the good result that will happen if society is sufficiently wary,
or something the courts do in order to keep society sufficiently wary, or maybe something else.
without the need for proof of mens rea
OK, now I think I see. If most offences require evidence of mens rea, then when a statute doesn't explicitly mention the need for such evidence, the courts will interpret the statute as requiring it implicitly. But if the law defines a large number of offences explicitly as not requiring evidence of mens rea, then courts will stop making that presumption, and it will be much easier for people to be convicted of crimes when they had no ill intent.
unless it specifies explicitly that that is what it is doing.
Yup, I correctedly figured out the meaning of the whole sentence before getting to this clause. It wasn't easy, though! Strictly speaking, this final clause is redundant, but it's still helpful because it confirms the reasonable interpretation I arrived at just before I read this clause.
Here's something very important, which contains an important lesson for how to interpret these complex sentences with lots of negations. The word "wary" suggests that a danger is coming: something to be avoided—so it's a kind of negation. However, the clause starting with "so" doesn't describe the danger, it describes the good result of being sufficiently wary. Naturally, though, the "so" clause describes the good result as a negation of the danger, otherwise you'd never know the point of the sentence.
The good result is that courts do not presume that they should convict people for offences without proof of mens rea. The good result includes convicting people for offences without proof of mens rea, but only when statutes explicitly direct the courts to do that. Stated affirmatively, the good result is that courts require proof of mens rea even when the relevant statute doesn't mention mens rea.
The lesson is: you must use common sense to infer what is to be negated and what is not, on the principle (the presumption!) that the author is reasonable. You presume that the sentence makes sense—is logical—and you interpret based on that assumption. This can only be done by considering the subject matter itself. You can't learn rules of symbol-manipulation that replace every term with a meaningless letter and grind out answers according to pristine rules, or you'll introduce nonsense that was not intended and is not a reasonable interpretation.