What example of fallacy is "If such actions were not illegal, then they would not be prohibited by the law"?

I have seen this exact example elsewhere as an example of begging the question, but I don't agree—at best, it is a poor example. It is a statement of a definition. It would be like saying, "If he is incapable of having children, then he is sterile." In your example, the conclusion does not assume the premise; the conclusion is the premise redefined. Begging the question has to be more of an argument than a definition. My "litmus" test is to us the same words in what might be the premise and the conclusion and if the meaning does not change, it is not a fallacy, but a definition. For example:

"If such actions were not prohibited by the law , then they would not be prohibited by the law"

Now let's look at a strong example of begging the question:

Paranormal activity is real because I have experienced what can only be described as paranormal activity.

In this second example, we are not defining paranormal activity, but we are making a claim that it is real. We are assuming the conclusion (that paranormal activity is real) based on the assumption being made in the premise (that I have experienced what can only be described as paranormal activity).

I would argue that it's either circular reasoning, or else ad nauseaum.

Consider you question: "If such actions were not illegal, then they would not be prohibited by the law." As noted by Bo above, the word "illegal" and the phrase "prohibited by law" have an identical concept behind them. I don't have the correct terms for it, but they both point to the same "symbol," if you will; to the same thing stripped of the linguistics used to convey it. You can abstract the statement to "If X [' such actions '] were false [' not illegal '], then X [' they ' {referring back to 'such actions'}] would be false [' not prohibited by law '] ," or cleaned up "If X were false, then X would be false.

If, however, you would make the claim circular reasoning needs two separate, interlocking sets of premises and conclusions, then I instead would say this is ad nauseaum. Blatantly repeating the same proposition, hoping that it is accepted.