Or, using your EOF markers, you need to quote the initial marker so expansion won't be done:
#-----v---v------
cat <<'EOF' >> brightup.sh
#!/bin/bash
curr=`cat /sys/class/backlight/intel_backlight/actual_brightness`
if [ $curr -lt 4477 ]; then
curr=$((curr+406));
echo $curr > /sys/class/backlight/intel_backlight/brightness;
fi
EOF
IHTH