This is an expression that is frequently used in legal or other kinds of official written texts. By including but not limited to a, b, and c is meant that a, b and c are not exclusive but representative of the wider group. It is used in order "to refer to something in a general way but also make specific mentions of certain things" (incorporated zone).
This set phrase is voluntarily redundant, and by that I mean that including a, b and c would be more than enough and correct to mean that other items are part of the wider group to which a, b and c belong. Lawyers, however, use this phrase keeping in mind that their opposition might accuse them of not being explicit enough. See for example what this law oriented site has to say about the matter:
Legal drafting is always subject to hostile misreadings by opposing parties who will argue that implications aren’t enough [in our case, the implication that no one will misunderstand "including" as introducing a non-exhaustive list; that it clearly implies that not only a, b and c are included, but other items, too].
In more simple words, this phrase is used as an over explicit overstatement, for "fear" of being accused of not covering other possible options that may be of legal consequence.
In your case, it is a bit more simple. This is not a legal statement, but is meant to sound as official and formal as can be. "Wrist-watch, cellphones, calculators" are not exclusive (not the only ones) but representative of the wider group of "personal items".
Think of your sentence in this way:
You will not be permitted to bring any personal items to the test centre, such as (but not only) wrist-watch, cellphones, calculators, etc.
Reading between the lines, you can understand:
If you come with an mp3 player and earphones (or something of the kind), do not tell us that it was not listed in the exam regulations.
We stated, and even overstated for that matter, that we did not
only mean wrist-watch, cellphones, calculators, but other such
personal objects, too. Therefore, by using the expression "including but not limited to" we are covered. If you didn't get it, that's your problem, brush up your English!
Now, your particular sentence does not mean that you should not bring ANY personal item, but that you should not bring any personal item of this kind. You will take a pen to an exam, and you will have clothes on and maybe an earring in your nose and the like. All these are personal items, too, but the examiners will not be concerned with that kind of personal items.
You can find plenty of other examples on Ludwig which will help you become more familiar with the use of this phrase. Here is a simple one:
We cover various topics including, but not limited to: business, technology, culture, media and the future. (meaning: these are some topics, but the list is longer, and we do not pretend we cover them ALL)
Note 1: Don't use the phrase if you don't want to sound pompous and overcorrect. Including is enough! In a legal or official context, including, but not limited to may be recommended or even required, but you'd better ask for advice in drafting declarations or statements.
Note 2: It is interesting and funny at the same time that your sentence states in three ways (!) that the list of personal items it gives, is not exhaustive:
- but not limited to
- , etc. (and others)
Looks like these guys really don't want to get busted for using "inexplicit or inaccurate language"!