Not only is that possible, but we do it all the time.  

 - No one can see the apps you register at [stackapps.com/apps/oauth/register][1] except you (and the SE devs).
 - **If you don't need OAuth**, then [you can enter just about any fake information in the required fields][2].  
  This still allows you to use the `key` to get 10K quota.

It is only when you want to access `private_info` or write to the site that you need OAuth and [a public post here][3].

See, also: https://stackapps.com/questions/4573/my-app-has-to-be-published-first-but-its-still-under-development

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**If you *do* need OAuth**, your app/script post here does not have to link to your domain ([if any][4]).  So the public need not know anything other than that you have an app.

But beware that [there has been a backlash against stagnant or malformed app/script listings][5].


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Finally, if you **do not need OAuth, and you make less than 300 API calls** per day, per IP address, then you do not need to register your app at all.  
The first and biggest benefit of registration is the increased quota.

  [1]: https://stackapps.com/apps/oauth/register
  [2]: https://stackapps.com/a/6406/7653
  [3]: https://stackapps.com/questions/7/how-to-list-your-application-library-wrapper-script-here
  [4]: https://stackapps.com/questions/7812/getting-an-api-key-for-an-application-without-a-server
  [5]: https://stackapps.com/questions/7490/what-to-do-about-the-tsunami-of-placeholder-apps-created-because-of-a-poorly-wri