I'm attempting to create my own atoi function. With the following I'm getting a return value of 0. Whatever I change the number variable within the function is what I get as a return value. Any suggestions on modifying the code?
//my atoi function
int atoi_me(char *numstring)
{
    int number = 0;
    while((*numstring >= '0') && (*numstring <= '9'))
    {
        number = (number * 10) + (*numstring - '0');
        numstring++;
    }
    return number;
}
int main()
{
    char *number[MAXSIZE];
    int num;
    printf("Please enter a number:\n");
    scanf("%c", &number);
    num = atoi_me(*number);
    printf("%d", num);
    return 0;
}
                You're declaring an array of char *, that is, an array of strings, rather than a single string.  You probably want: 
char number[MAXSIZE];
Your scanf format string is wrong.  If you want to read a string, you should use %s.  %c reads only a single character.
Your scanf parameter is wrong - pass number itself (or &number[0] if you prefer), not &number.
The parameter you're passing to atoi_me is wrong.  Call it with number (or equivalently &number[0]), not *number.
Putting all of that together, you should have a main routine something like this:
int main(void)
{
    char number[MAXSIZE];
    int num;
    printf("Please enter a number: ");
    scanf("%s", number);
    num = atoi_me(number);
    printf("%d\n", num);
    return 0;
} 
Editorial notes: You have a potential buffer overflow with the scanf line.  You'd be better off using a function like fgets(3) that makes it easy to protect against that kind of problem.
atoi(3) also traditionally supports negative numbers (with a leading -) and an optional leading + for positive numbers, which your implementation doesn't handle.
As I thought, the problem is in your call.
Change your main to.
int main()
{
    char number[MAXSIZE];
    int num;
    printf("Please enter a number:\n");
    scanf("%s", number);
    num = atoi_me(number);
    printf("%d", num);
    return 0;
}
Other than this it's not a good idea to use scanf - http://c-faq.com/stdio/scanfprobs.html . In this case you should use fgets.
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