App updates show as a number on the Store Tile for 24 - 48 hours. Store checks for updates every 24 hours, and if a data connection is not available, the update will not run again for 24 hours. Try the following: 

1. Make sure that you have the date and time set correctly on your phone (Settings > date+time).

2. Check your internet connection: Open the web browser and check if you can access Store by entering the address in your browser address field.

3. Tap and hold the Store tile, and tap the unpin icon. Then go to the application list and re-pin the Store app to the start screen.

4. Make sure that you have authorised Microsoft to send messages to your number for Store notifications. Use a desktop browser, sign in to your Windows Phone account using the phone's primary Microsoft (Live) account ID, and then select Update on the left side.

5. Go to Settings > find my phone, and then check the Send apps to my phone using push notifications (not SMS) checkbox.

6. Turn off at least one app with background tasks enabled (the app update notification may not be shown if the background tasks exceed a certain limit (around 15).

  1. On the start screen, swipe left, and select Settings.
  2. From settings swipe left to applications tab, and select background tasks.
  3. Tap on the installed app you want to modify, and then select block.
  4. Tap the Back key to return to the start screen.
  5. Unpin the app from the start screen.