I assume you can't use StringBuffer and you can't use StringBuilder either, as this wouldn't make sense otherwise.
You can concat two Strings after converting them to a char array:
public String concat(String[] args){
int length = 0;
for (String s : args) {
length += s.length();
}
char[] result = new char[length];
int i = 0;
for (String s : args) {
s.getChars(0, s.length(), result, i);
i += s.length();
}
return new String(result);
}
Here's how you test it:
public void testConcat() {
String[] s = new String[3];
s[0] = "abc";
s[1] = "def";
s[2] = "ghi";
System.out.print(concat(s));
/*
* Output: abcdefghi
*/
}