How to store an array of an unknown amount of strings of unknown length from input?

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I have trouble with storing strings in an array from input. I want to store it from input ending with EOF and be able to store any amount of strings of any length. Here is my code:

char **list = (char**)malloc(sizeof(char*));
char c = getchar();
int i, j, count = 0;
int size = 0;
char * string = NULL
while (c != EOF)
{
    size = 0;
    while (c != EOF && c != '\n')
    {
        string = (char*)realloc(string,size+1); //line 210
        string[size] = c;
        size++;
        c = getchar();
    }
    list = (char**)realloc(list, (count+1)*sizeof(char*));
    list[count] = string;
    ++count;
    string = NULL;
    c = getchar();
}
for (j = 0; j < count; ++j) //trying to print out all the strings
{
    printf("%s\n", list[j]); //line 237
}
free(string);
free(list);
return (EXIT_SUCCESS);

I know I should also be testing whether all the allocs where successful, but that is not my issue right now. The program works properly when compiled with gcc -Wall -pedantic, but valgrind gives the following errors:

==2601== Memcheck, a memory error detector
==2601== Copyright (C) 2002-2015, and GNU GPL'd, by Julian Seward et al.
==2601== Using Valgrind-3.11.0 and LibVEX; rerun with -h for copyright info
==2601== Command: ./a.out -g
==2601== 
asd
dsa
==2601== Invalid read of size 1
==2601==    at 0x4C30F74: strlen (in /usr/lib/valgrind/vgpreload_memcheck-amd64-linux.so)
==2601==    by 0x4EA95DB: puts (ioputs.c:35)
==2601==    by 0x4008E8: main (testing.c:237)
==2601==  Address 0x52035c3 is 0 bytes after a block of size 3 alloc'd
==2601==    at 0x4C2FD5F: realloc (in /usr/lib/valgrind/vgpreload_memcheck-amd64-linux.so)
==2601==    by 0x40080D: main (testing.c:210)
==2601== 
asd
dsa
==2601== 
==2601== HEAP SUMMARY:
==2601==     in use at exit: 12 bytes in 4 blocks
==2601==   total heap usage: 14 allocs, 10 frees, 2,101 bytes allocated
==2601== 
==2601== LEAK SUMMARY:
==2601==    definitely lost: 12 bytes in 4 blocks
==2601==    indirectly lost: 0 bytes in 0 blocks
==2601==      possibly lost: 0 bytes in 0 blocks
==2601==    still reachable: 0 bytes in 0 blocks
==2601==         suppressed: 0 bytes in 0 blocks
==2601== Rerun with --leak-check=full to see details of leaked memory
==2601== 
==2601== For counts of detected and suppressed errors, rerun with: -v
==2601== ERROR SUMMARY: 2 errors from 1 contexts (suppressed: 0 from 0)
1

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ad absurdum On BEST ANSWER

Instead of freeing string at the end of the program, you need to free the memory allocated for each of the strings that you read:

for (j = 0; j < count; j++)    
        free(list[j]);
free(list);

To fix the "Invalid read of size 1" error, you need to change:

string = realloc(string,size+2);

You are using size as the index for string, so you need to increment this value before adding a space for the '\0' in the allocation. Better variable names would have helped avoid this issue. Consider changing size to index.

Note also that you need c to be an int in order to store the EOF character, so change:

int c = getchar();

Also, this code is failing to terminate the strings with a NUL. You need to add:

...
string[size] = '\0';    // NUL terminate string

list = realloc(list, (count+1)*sizeof(*list));
...

When you initialize list at the beginning of the program, there is no need to allocate space yet, and you just end up reallocing it anyway, so:

char **list = NULL;

And notice that I removed the casts from the calls to malloc() and realloc(). These are not needed in C, and mostly serve to clutter the code. Similarly, instead of using sizeof(char*) here, I used sizeof(*list), which is less error-prone and much more clear.