Seafile

Please note:

1. Seafile was a company with two branches - one in Germany (Seafile.de) and one in China (Seafile.com). In the summer of 2016, the two parted company.
2. Since then until now (March 2017), Seafile.de continued to provide a service syncing people's files using the open source Seafile client. Very good prices, very nice service, and an excellent client - one of the very few I've found able to deal with (literally) hundreds of thousands of small data files without breaking down. Great upload/download speeds. (I live in the EU, but not in Germany.)
3. As of March 2017 I noticed a rebranding of Seafile.de to Syncwerk.com (they Seafile.de redirects there now). I think Seafile GmbH (the German company) are seemingly going for a rebrand to move towards a new future. Good luck to them. I'm on board as a paying customer. Let's see where it leads. I hope they maintain the open source nature of their client.

If you don't know: Seafile.de (now SyncWerk.com) are a very reliable, low-price alternative for file syncing of the Dropbox type. Additionally, their client allows encryptions of the files you store. And the files are stored on EU territory. All of that is hugely important. By contrast, Dropbox are closed source, unencrypted (they encrypt only server-side, which is to say they can read your data) and they are US-based, which means they must hand over data at the NSA's behest without even telling you (see National Security Letter). So go with Seafile/SyncWerk. Or Tresorit. Or SpiderOak. But at least someone who treats your data with the decency and privacy you deserve.

Generous free version. Has some bugs in the iOS app.

Not a bad program for storing files. Utilities for this PC software product are not bad enough. Has a bunch of bugs and problems that I have not been able to solve personally. If you're a technically savvy person, try it out, you might like it. I didn't like this product, so I went to other analogs.

Unlike Nextcloud it doesn't provide additional functionality like Calendars, Contacts and Notes. It does file sharing much better and faster though!

I can connect to Amazon S3 for cheap storage, access my files as if they were on an external drive, and access my files on Android.

Very clean, good desktop integration, free and open source (self-hosted).

Seafile should be congratulated for offering optional client-side encryption of a folder you might want to sync. This is good because the files are encrypted on your computer before they are uploaded, which means that they are not readable to Seafile or anyone who might access their servers (legally or illegally). This makes your data more secure and private than you get with Dropbox, GoogleDrive, Box or OneDrive.

However...

You should know that the encryption the Seafile software provides encrypts the file contents only and not the name, file size, folder structure or the date any particular file was modified. This is a limitation if you want to prevent anyone learning what files you have, even if they can't read the contents.

Therefore...

A good solution is to pair Seafile's service with an open-source encryption program like Cryptomator. The reason for this is that Cryptomator provides a very simple way to encrypt your data on your computer before it syncs, but it does encrypt the file names, folder structure and other metadata. Cryptomator is also preferable for this purpose over competitors like BoxCryptor because it is free and open source, and therefore its code can be inspected to ensure that the encryption is implemented correctly.