If you're looking for pre-indieweb / legacy feed readers (e.g. like Google Reader) see:
Reasons to have a personal reader (a reader integrated into your site for you to use)
That's just the beginnings of what an indieweb reader can enable.
In datetime order of implementation (earliest first)
acegiak developed a Wordpress plugin, WhisperFollow, which aggregates RSS/Atom and Microformats2 data from the pages linked to in your blogroll and displays those updates in a private page in your wordpress blog. It currently defaults to RSS/Atom only attempting to read MF2 if the separate feed isn't found. It also allows updates to be reblogged and commented on.
Barnaby Walters developed Intertubes, an experimental indieweb-oriented flow-based programming + feed reader UI parsing microformats and shimmed twitter.com microformats.
Barnaby is using and developing Shrewdness, a multi-column feed reader as of at least 2014-08-04
See http://andysylvester.com/2014/03/01/howto-setting-up-the-selfoss-feed-reader-with-microformats-support/ for instructions on how to set up the Selfoss reader.
Ben Werdmüller, Aaron Parecki, Emma Kuo
Malcolm has added microformats support to his version of the SimplePie feed parsing library, which is available at https://github.com/mblaney/simplepie. It is deployed at unicyclic.com which includes:
Aaron Parecki developed a Microformats2 plugin for selfoss, and ran it on a subdomain of aaronparecki.com since 2014-02-13. The reader polls the subscriptions at a predefined interval looking for new h-entry posts on each person's home page.
Here's a screenshot of a following list:
Here's a screenshot of a reading window:
For more on this see: http://aaronparecki.com/notes/2014/02/13/3/microformats-selfoss-indieweb
Integrated Activity Reading
From analyzing the screenshots in this article:
of Facebook's News Feed experiment:
A single integrated news feed of friends' activities, including posts and likes.
Comparing the experiment and the reversion, shows the experiment provided:
All of these differences provided for a better-for-the-user user experience (faster, more focused, more relaxed) which could (should) be used in the design of indieweb reader user interfaces, as it provides opportunities to outdo silo UX.
Tantek 16:36, 27 March 2014 (PDT)
Jeena Notes Reader
Thoughts on challenges and how to develop an indie notes reader:
Shane Feed Me See More
I (Shane) have "plans" to build a personal reader, as a separate project, called Feed Me, See More. I made a static prototype in 2010-08 (with static snapshots of real data from then)
The plan is for Feed Me See More and Homesteading to be sister projects. Subscribe webactions (while browsing around the web) will get caught by Feed Me See More. Repost/Reply/Bookmark webactions will go from my Feed Me See More app to my Homesteading powered site. But both will be optional connections to the other. Either Feed Me See More or Homesteading could be run stand-alone without the other one. At least, that's the Big Plan™. (Not enough time/money/people.) -Shane (2013-08-10)
Autodiscovery of h-feeds
List and hashtag
In 2015, at the XOXO festival, Aaron Parecki created a private stream of tweets from attendees who mentioned "xoxo," and those tweets appeared in the conference's Slack channel. This was hugely popular at the conference since the hashtag on Twitter was being spammed, making it impossible to read actual content about the conference on Twitter itself.
In-stream comments and likes
As discussed on IRC 
I enjoy the integrated experience of an indie reader, but am missing comments and likes from others. This is especially noticeable when I want to comment on something and then realize someone may have made the same comment, a better comment, or a comment that makes my comment seem silly.
A "(n) comments" indicator would be a nice alternative/addition.
As Twitter, Tumblr, and other aggregators have demonstrated, keeping up with everything you've subscribed to is both challenging, and eventually unrewarding. Thus some amount of smart filtering/pruning/batching is likely to be an essential part of making indie readers scale beyond following only a handful of people. See also:
Whoever is working on the indie reader problem, please consider the problem of social media distraction (which seems to come more from the reading side than the writing side). See related: silo-quits.
missing good stuff
Likely as an attempt to compensate for #overload as noted above, users tend to limit the number of sources they follow, thus resulting in another challenge - the loss of signal, AKA "missing good stuff".
It is possible that there are some approaches for mitigating overload (e.g. perhaps categorizing notes with non-self-source links as comments on those links, and aggregating them by link destination) that could help bubble up "good stuff" from a larger set of people you follow.
One of the reasons I have not prioritized the reader for myself, because I have better luck reading social media by searching for hashtags (results piped to IRC) and having other people send me things :)- Aaronpk in IRC.
It becomes more and more time consuming to maintain a large list of subscriptions in traditional feed readers. It isn't uncommon to see feed URLs go dark, change location or just stop working for whatever reason. It would be advantageous for the next crop of readers to provide tools to help automate this sort of maintenance. It would also be great to find ways for feeds to state "This will never be updated again" and "This feed will be moving to this new location".
Open source examples
Many social content hosting silos have integrated reader like features, but only for "feeds" of accounts on the particular silo.
After implementing an indie reader, you may find that you want to search all the things you read. See: