[redhead]

Can I do things like this with pointers to pointers?

The way you're using it there, No you cant.
The suggested function isn't valid.. And I think you know that too..
First off, you're using strcpy() which is provided in string.h, but you don't include it.
Second, you're accessing placeholder at index 0 of youre string_table struct pointer, this is an invalid assignment, since you crap on your orriginal deffinition of string_table
This is what I would do in that case:

/* somefile.h. */
#ifndef SOMEFILE__
#define SOMEFILE__
#include <stdio.h>
#include <stdlib.h>
#include <string.h>

typedef struct
{
    char **string_table;
    int num_strings;
} string_table;

int add_str_to_table(string_table*, char*);

#endif

/* somefile.c */
#include "somefile.h"

int add_str_to_table(string_table* tbl, char* str)
{
  tbl->string_table = (char**)realloc(tbl->string_table,
                                      (tbl->num_strings +1)*sizeof(char*));
  if(!tbl->string_table){
    printf("Not enough heap space to reallocate string_table\n");
    return -1;
  }
  tbl->string_table[tbl->num_strings] = (char*)malloc(strlen(str));
  if(!tbl->string_table[tbl->num_strings]){
    printf("Not enough heap space to allocate added string\n");
    return -2;
  }
  if(!strcpy(tbl->string_table[tbl->num_strings], str)){
    printf("Unable to copy str to placeholder\n");
    return -3;
  }
  tbl->num_strings++;
  return 0;
}

int main()
{
  string_table * my_tbl;
  int i;
  char str1[] = "This is a string";
  char str2[] = "This is a second string";
  my_tbl = (string_table*)malloc(sizeof(string_table));
  my_tbl->num_strings = 0;
  if(0 != add_str_to_table(my_tbl, str1)){
    printf("Unable to add str1\n");
    return -1;
  }
  if(0 != add_str_to_table(my_tbl, str2)){
    printf("Unable to add str2\n");
    return -1;
  }
  for(i=0; i < my_tbl->num_strings; ++i)
    printf("my_tbl->string_table[%d] = %s\n", i, my_tbl->string_table[i]);
  return 0;
}