Find most frequent word in text file
Clearly employ contract requirements

"Write a function that ...  contains unknown number of words of max length 10 each." should echo in code someplace a 10 named something like "words of max length".  Use "size" for the size of the array.  With strings, best to use "length" in the context of the number of characters before the '\0'.

//#define MAX_CHARS 11
#define WORD_MAX_LENGTH 10 
#define WORD_SIZE (WORD_MAX_LENGTH + 1)
... 
// calloc(MAX_CHARS, ...
calloc(WORD_SIZE, ...


Use int with fgetc()

fgetc() returns values in the unsigned char range and the negative EOF.  These typical 257 different values will not distinctively fit in an 8-bit char.  With original code, should char be an unsigned char, your loop is infinite.

// char ch;
int ch;
...
while((ch = fgetc(file)) != EOF) {


Check return value of fopen() and calloc()

Should the open fail, detect it.  

FILE *file = fopen(filename, "r");
if (file == NULL) {
  fprintf(stderr, "Unable to open file &lt;%s&gt;\n", filename);
  return 0;
}


Should the allocation of more than 0 bytes returnNULL,detect it.  

arr = (char**) calloc(counter, sizeof(char*));
if (arr == NULL &amp;&amp; counter != 0) {
  fprintf(stderr, "Out of memory\n");
  return 0;
}


Improve *alloc() style

Cast not needed.
Allocate to the size of the object, not the type.  Easier to code right, review and  maintain.

// arr = (char**) calloc(counter, sizeof(char*));
arr = calloc(counter, sizeof *arr);


Good use of sentinels in printing strings

 //                                v   v
 printf("The most frequent word is \"%s\" ...


Lack of bounds checking

Code should ensure i, j are not to big.

if (i &gt;= counter) Handle_Error_Somehow();
i++;

if (j &gt;= MAX_CHARS) Handle_Error_Somehow();
else { 
  arr[i][j] = ch;
}


Missing fclose()

fclose(file);




Little bits.

I'd expect size_t counter, i, j, k for array indexing as size_t is neither too narrow nor too wide.  Be mindful it is an unsigned type.

Style: if (index_max != NULL){ is replaceable with if (index_max) {.  I find the the 2nd clearer.