I am automatically updating (and deploying) a web app in GAS, and the project version number is already over 270. I am worried there might exist an upper limit, but I cannot find any information regarding this.
There is a limit of 200 versions per script, as per the official Quotas for Google Services page. To view the number of versions you currently have on a Google Apps Script (GAS) project, go to the Project History tab of your GAS project and on the bottom right of your screen press the trash icon. The top of the modal menu that appears shows the number of versions you currently have on your script, as well as the versions of your GAS project that are currently in use.
Versions of a GAS project can be deleted by using the Project History tab.
To delete just one version, go to the Project History tab and on the right of your screen find the version you would like to delete. Select the kebab menu that shows up when you hover over the version and press Delete this version.
To delete multiple versions at once, go to the Project History tab and on the bottom right of your screen press the trash icon. Select all of the versions you would like to delete in the modal that appears and press the Delete button.