Question: I would like to be able to play a piano or keyboard and have what I play stored and reproduced as musical notation. Is that possible?

Answer: 
This solution might involve a bit of a learning curve, but there is some software called [Lilypond](https://lilypond.org/) for typesetting music-notation, and it has a GUI frontend program called [Denemo](http://denemo.org/). In Denemo you can play midi into it and generate beautiful/professional scores. Lilypond/Denemo have an intimidating number of features.

Here are a couple of demo videos of someone using a midi-keyboard to enter notes:

[http://www.denemo.org/~rshann/PitchesFirst.mp4](http://www.denemo.org/%7Ershann/PitchesFirst.mp4)

<https://vimeo.com/62188678>

Caveats:

* I've used Lilypond but haven't actually tried Denemo
* The workflow is a bit clunky-looking
* I am not affiliated with either of these programs in any way.

Other software with similar features:

* Finale can import MIDI files but you can't type in MIDI directly.
* I believe even the free version of Avid's Sibelius software *does* support entering MIDI directly.
If you expect the process to be fully automatic and the music you play is anything above the simplest - then NO it's not possible. You can get an exact reproduction of what you play, but you will not get even semi-professional results with the score notation.