Or will you be required to give up one of the other two citizenships before you are able to gain it?
Yes, you can have three or more citizenships as a South African citizen, so long as:
- You have applied for retention of South African citizenship before each acquisition of citizenship.
- The other countries that you are a citizen of allow multiple citizenship.
It is common for websites and government documents to refer to dual citizenship (implying a limit of two), while they should instead be referring to multiple citizenship.
As per other answers to this question, Australian citizens gaining a different citizenship are allowed to keep their citizenship - see http://www.citizenship.gov.au/current/dual_citizenship/
Note that although the title refers to dual citizenship, the body of the page refers to "two or more countries".
New Zealand citizens are similarly allowed to have multiple citizenships - see the wikipedia article at http://en.wikipedia.org/wiki/New_Zealand_nationality_law#Dual_citizenship
Again, it's worth noting that although the Wikipedia article heading refers to dual citizenship, the body of the Wikipedia text actually refers to multiple citizenships.
As mentioned in the comments on this question, unless a South African applies for "Retention of South African citizenship", they will lose their South African citizenship on gaining any other citizenship.
Dual vs Triple Citizenship
Although the South African high commission in Australia web pages refers to dual citizenship, the South African Citizenship Act, 1995 (see Chapter 3) makes no reference to any specific count of citizenships, and does not contain the word 'dual'.
The legislation only refers to the acquisition of citizenship, rather than referring to 'a second citizenship', a 'third citizenship', or similar.
The Retention of South African Citizenship Form also makes it quite clear that you can apply for a third citizenship.
Question 10 asks you to list any citizenship you already have, in addition to your South African citizenship. This means that you can use the form to apply for a third citizenship.
With reference to the original question, you would need to fill in the form as follows:
- Question 9: You would fill in that you are acquiring Australian citizenship.
- Question 10: You would fill in details of your existing New Zealand citizenship.
It is unclear form the form how one would apply for retention of a fourth citizenship. I expect it would simply be written into the space allocated for Question 10.
According to Dual South African Citizenship, permission is required before acquiring another citizenship:
If a South African citizen does not obtain this prior permission they will automatically lose their South African citizenship on voluntary acquisition of a foreign citizenship.
There is no such restriction for New Zealand citizens. Australia recognises holders of dual (or more) citizenships without problems (from Dual citizenship):
Australia allows its citizens to hold dual nationality.
So, one would need to obtain permission from the South African government before undertaking application for Australian citizenship.