I think that
(.).*\1
would work:
- (.)match a single character and capture
- .*match any intervening characters
- \1match the captured group again.
(You'd need to compile with the DOTALL flag, or replace . with [\s\S] or similar if the string contains characters not ordinarily matched by .)
and if you want to require that it is found at least 3 times, just change the quantifier of the second two bullets:
(.)(.*\1){2}
etc.
This is going to be pretty inefficient, though, because it's going to have to do the "search for the next matching character" between every character in the string and the end of the string, making it at least quadratic.
You might be as well off not using regular expressions, e.g.
char[] cs = str.toCharArray();
Arrays.sort(cs);
int n = numOccurrencesRequired - 1;
for (int i = n; i < cs.length; ++i) {
  boolean allSame = true;
  for (int j = 1; j <= n && allSame; ++j) {
    allSame = cs[i] == cs[i - j];
  }
  if (allSame) return true;
}
return false;
This sorts all of the same characters together, allowing you just to pass over the string once looking for adjacent equal characters.
Note that this doesn't quite work for any symbol: it will split up multi-char codepoints like . You can adapt the code above to work with codepoints, rather than chars.