Valgrind jump or move depends on uninitialised value(s) using getline and one char *

You are close, but your use of current_size += result; before the call to realloc() prevents a simple check of current_size == 0 to check for the first iteration which you would then want to initialize final as an empty-string.

You can do the following:

#include <stdio.h>
#include <stdlib.h>
#include <string.h>

int main (void) {

    char *final = NULL, *line = NULL;
    size_t n = 0, current_size = 0;
    ssize_t result;
    
    while ((result = getline(&line, &n, stdin)) != -1) {
        if (result > 0)
            line[--result] = 0;                                 /* trim \n */
        
        final = realloc (final, current_size + result + 1);     /* add room for \0 */
        if (final== NULL)                                       /* validate allocation */
            return 1;
        
        if (!current_size)                                      /* if 1st word */
            *final = 0;                                         /* make final empty-str */
        
        current_size += result;                                 /* now update value */
        strcat (final, line);                                   /* concatenate */
    }
    
    printf ("%zu - %s\n", current_size, final);
    
    free (final);
    free (line);
}

Example Use/Output

$ ./bin/getlineplusone << 'eof'
1234
56
789
eof
9 - 123456789

Memory Use/Error Check

$ valgrind ./bin/getlineplusone << 'eof'
1234
56
789
eof
==8438== Memcheck, a memory error detector
==8438== Copyright (C) 2002-2017, and GNU GPL'd, by Julian Seward et al.
==8438== Using Valgrind-3.13.0 and LibVEX; rerun with -h for copyright info
==8438== Command: ./bin/getlineplusone
==8438==
9 - 123456789
==8438==
==8438== HEAP SUMMARY:
==8438==     in use at exit: 0 bytes in 0 blocks
==8438==   total heap usage: 6 allocs, 6 frees, 5,262 bytes allocated
==8438==
==8438== All heap blocks were freed -- no leaks are possible
==8438==
==8438== For counts of detected and suppressed errors, rerun with: -v
==8438== ERROR SUMMARY: 0 errors from 0 contexts (suppressed: 0 from 0)