Logos is recognized as having the largest digital library for Biblical and Theological studies. Most people don't realize that Logos as a software is itself free - just go to the download page and download it.
Furthermore, creating a Logos account is free as well – again, just go to Logos.com and create an account. The log in credentials you create is what you use to sign in to the app.
While Logos is a business that is out to make money, they do provide some great free content as well. In fact, even a casual user can get some fairly good content to do some basic Bible study. What I want to do in the remainder of this post is highlight this free content, particularly the resources that would be of interest to academics.
First, you can see a full list of everything that costs $0 by going to this page. The list below will highlight the most important ones.
Faith life Study Bible. Not only is this one of the best study bibles out there, but it is much more than a study bible. It includes videos, infographics, tables, and timelines.
Lexham Bible Dictionary. This is destined, I think, to be the premiere (mid-sized articles) Bible dictionary. It is already huge, and continues to expand.
Lexham English Bible (with audio). The LEB is a fantastic translation. It is a more word for word translation, so it does not necessarily always read flowingly, but is great for study. You can also grab the free audio version of the ESV here.
SBL Edition of the Greek New Testament. This is a fully-tagged and fully searchable version of the Greek NT. It includes the apparatus.
Perseus Classics Collection. The entire Perseus collection, over 1,000 volumes, tagged and searchable in Logos. This covers (I think) all of the ancient classical Greek and Latin literature. Because the collection is so big, the parsing is not always accurate. I'm not sure if Logos is working to correct this or not. Nonetheless, this is invaluable.
Duke Database of Documentary Papyri. Part of the larger Perseus collection, this contains 256 volumes of primary texts in Greek, Latin, Coptic, and other languages.
Codex Bezae. One of the most important uncial manuscripts in transcribed form. Dated to the 5th century.
Codex Sinaiticus. One of the most important manuscripts in transcribed form, dated to the 4th century.
Books.logos.com. A new beta program for Logos is a bit of a google books type project. They are digitizing some old works from particular libraries, and there certainly are some gems there. There are currently some limitations on these books – for instance it isn't easy to browse the books on the site right now. The biggest downside that I sincerely hope is remedied is that these are treated in Logos as media collections - in other words every page is treated as an image. This means they aren't searchable. As I said, I hope Logos runs OCR on these and makes them searchable in the future. Also, because these are image collections, it takes up more space than other resources. For primary literature, here are a few gems:
Migne's Patrologiae Graeca
The Oxyrhynchus papyri volumes
Go forth, download, and enjoy!