There is the Google Play Reader. I haven't used it, but I did install it now to see if you could read your own books in it.
https://play.google.com/books/uploads
Quote:
Upload your personal documents to Google Play, and read them anywhere.
Keep up to 1,000 EPUB or PDF files in Google Play for free.
Upload from your computer or import from Google Drive.
Read anywhere: in the web browser, on Android phones and tablets, or with iPhone, iPad, and iPod touch.
Page position, bookmarks, and notes are synced to the cloud, so you can pick up reading where you left off.
File size can't exceed 50MB, and it may take a few minutes to finish processing before you can read.
Make sure you have the appropriate rights to the uploaded content.
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(I clicked on "SHOP" in the left hand menu, and then "MY BOOKS" to get the upload menu. Even though I start in MY BOOKS, the upload options weren't there until after I went to the SHOP.)