(This is a wiki-style post. You can update the list by editing this post.)
Below is a wonderful list of “Google alternatives” compiled by Christian Pietsch and others.
Eventually, we want to have separate topics for each of these products/services under either the Alternatives category or the Stopgaps category so we can discuss them individually.
I’ll get on that soon, but for now, this is a Wiki post, so anyone can edit it 
Huge thanks for Christian for all his hard work:
There are several reasons why you probably do not want to share too many aspects of your life with Google: It has a near monopoly in several aspects of the Internet, and has its hands in many others: search, advertisement, calendaring, cloud storage, social media, news, e-mail, maps, online office … This stfles competition. And by using several of these, you give Google the chance to gain new insights about you by combining data from several services.
Analytics
Blogger (blogging)
Self-hosted:
Calendar
Self-hosted:
Cloud-hosted:
Code (Version Control)
(Google Code is being discontinued anway) (version control is not necessarily limited to software development projects, it can also be beneficial to track changes to collaboratively developed documents or data)
Collaboration/Project Management
(Which Google service does this address?)
Docs (authoring)
Collaborative authoring:
Markup-based:
Offline authoring:
Chat/Hangout (IM/Video chat)
-
https://jitsi.org/ - End-to-end encryption through other services (gchat, facebook, etc.), encrypted IM/video chat using XMPP, in-browser video conferencing
Mail (e-mail)
Webmail (includes hosted IMAP server):
Self-hosted:
Client software:
Maps
Plus (social networking)
Reader (RSS)
Feedbin. The cloud version isn't free (I consider this a benefit, but not everyone will). It's open source though, and you can host it yourself.
Self-hosted:
Cloud-hosted:
Android-only:
Scholar (academic search)
Search (general)
Storage
Self-hosted (buy a Raspberry Pi or similar!):
Cloud-hosted:
Offline:
Advertising / Affiliate Network
Closing words
PS: Sorry if this list is a little bit Germany-biased at the moment. Please add more to eliminate this bias!
More suggestions for the privacy and security conscious: https://prism-break.org/
License for the contents of this page: CC-0 (https://creativecommons.org/publicdomain/zero/1.0/)