1. Titanium Backup
   

   This app allows you to create backups of your apps and their data, including system apps and settings. It's especially useful for those who frequently flash custom ROMs, as it allows you to quickly restore your apps and data after flashing a new ROM.

2. Greenify

   This app helps to improve battery life by identifying and hibernating apps that are using up too much background power. Greenify works by putting apps into a state of hibernation, which stops them from running in the background and consuming battery life.

3. Tasker

   This app allows you to automate a wide range of tasks on your device. It can be used to automatically change settings based on certain conditions, such as turning on Wi-Fi when you're at home, or turning on silent mode at night.

4. Xposed Framework

   This app allows you to install other apps, called "modules", that can add new features to your device or change the way existing features work. There are a wide variety of modules available, including ones that can change the look of your device or add new functionality to the Android operating system.

5. ROM Manager

   This app is specially designed for those who frequently flash custom ROMs. It allows you to easily manage and organize your ROMs, as well as to perform backups and restore your device.

Keep in mind, rooted devices can be vulnerable to malicious software and have less security guarantee, hence it is important to install apps from reputed sources and check reviews and ratings before installation.

Final Notes

This is not an exhaustive list of all the apps available for rooted mobile devices, but these are some of the most popular and useful ones. Each of these apps can add valuable functionality to your rooted device and enhance your overall experience.