1234 - C (56 characters)

[KernelEquinox]

Here ya go~

Compiles with gcc main.c

// a=argc (1)
main(a)
{
    // b=1, c=4
    // Loop until a == 5
    for(int b=a,c=b+b<<b;a<=c;)

        // c-b<<c  ==  3<<4  ==  48 (ASCII '0')
        // Add a to print its ASCII value
        putchar(c-b<<c|a++);
}

Minimal version:

main(a){for(int b=a,c=b+b<<b;a<=c;)putchar(c-b<<c|a++);}

[ISSOtm]

Clever one! However, it relies on the way it's called - only one argument, which puts it in the troll category. :/

[KernelEquinox]

Here we go, updated the code to be in accordance with the rules, 61 chars total.

main()
{
    // a=1, b=1, c=4
    // Loop until a == 5
    for(int a=!!a,b=a,c=b+b<<b;a<=c;)

        // c-b<<c  ==  3<<4  ==  48 (ASCII '0')
        // Add a to print its ASCII value
        putchar(c-b<<c|a++);
}

Minimal version:

main(){for(int a=!!a,b=a,c=b+b<<b;a<=c;)putchar(c-b<<c|a++);}

[ISSOtm]

Purrfect.