Error when running ((x++)) as root

In the absence of a hashbang, /bin/sh is likely being used. Some POSIX shells do support the ++ and -- operators, and ((...)) for arithmetic evaluations, but are not required to.

Since you have not included a hashbang in your example I will assume you are not using one and therefore your script is likely running in a POSIX shell that does not support said operator. Such a shell would interpret ((age++)) as the age++ command being run inside two nested sub-shells.

When you run it as a "normal" user it is likely being interpreted by bash or another shell that does support said operator and ((...)).

Related: Which shell interpreter runs a script with no shebang?

To fix this you can add a hashbang to your script:

#!/bin/bash
age=0
((age++))

Note: You do not need to terminate lines with ; in bash/shell.

---

To make your script portable to all POSIX shells you can use the following syntax:

age=$((age + 1))
age=$((age += 1))