Save the last successful backup and add --last-backup

Write the completion time and backup directory to
$XDG_STATE_HOME/simple_backup/last_backup after each successful backup
(as the target user when run as root), for status bar reminders.
--last-backup prints it.

README.md

@@ -62,6 +62,17 @@ The number of old backups to keep can be set with the --keep (or -k) argument (o
 Old backups are only removed if rsync completes successfully, so that they are not deleted when the new backup may be incomplete.
 The only rsync error that is ignored is return code 24 (source files vanished during the transfer), which is common when backing up a live system.
 
+## Last backup
+After each successful backup, simple_backup saves its time and directory to `~/.local/state/simple_backup/last_backup` (or `$XDG_STATE_HOME/simple_backup/last_backup`).
+When run with sudo or doas, the file goes to the home directory of the user who ran the command.
+Print it with:
+
+```bash
+simple_backup --last-backup
+```
+
+The file has two lines: the completion time in ISO 8601 format (e.g. `2026-10-25T14:03:12+02:00`) and the backup directory, so scripts (e.g. a status bar indicator) can easily read it to remind you when a backup is overdue.
+
 ## Remote backup
 > **Warning**
 > This feature is experimental