EE does not prevent users from re-registering onto an event.
It would be possible to do if you use the WP User Intergration Add-on but will require some custom coding from a developer if you want to prevent the users from being able to register onto the event again.
Here is an example of how you can display a message on the event page if the user has already registered that you can use a starting point:
The support post ‘One ticket per event’ is closed to new replies.
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Support forum for Event Espresso 3 and Event Espresso 4.