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Note that https://instavideosdownloader.com no longer really works so I've switched to using https://ingramer.com/downloader/instagram/ as an alternative, I'll have to see about editing those original posts.
I use the Firefox element inspector to upload photos and videos from Instagram. It is the most practical if you know a little HTML, you can copy the URLs directly, without downloading anything. I also use the Firefox extension Instagram-guest, it removes the login window.
To use this video as an example, here's the 480x600 version that plays normally:
http://s000.tinyupload.com/?file_id=26904739691389717631
And here's the clearer 640x800 version I get using the right download site (it's still a direct download from Instagram):
http://s000.tinyupload.com/?file_id=05618274878741103779
Not sure if there's a way to download a HQ version from mult-image posts or if one even exists, I'll have to try some other add-ons or something.
EDIT: Replaced the direct instagram example links with reuploads that won't expire, and a better example vid.
UPDATE: After some recent changes by Instagram that broke the other methods, the site https://instavideosdownloader.com is one of the only options left that seems to work somewhat reliably, as it gets alternate versions from everything else that are at a lower bitrate, but a lot of times are the only higher-res ones available. They're usually 720x720 or 720x900 instead. I recommend comparing them to any other versions fullscreen to confirm which are actually higher quality overall, whichever is the larger file is usually the actual best one, even if it's not the highest resolution (to reiterate, if instavideosdownloader gives you say a 640x640 video but you can also download a 640x640 video using another method, the latter will usually be a larger filesize and better quality, I've also seen where a lower-resolution version from other sources is still a larger file and better quality than the higher-res version from instavideosdownloader). It also seems to use alternate resized versions for still photos in some cases, so it's probably not the best option for those.
NEW UPDATE: The Instagram Video Downloader Firefox add-on by lumieras seems have stopped working around 10/26/19. This Instagram Downloader addon seems to be a good replacement for it, and handles multi-image galleries better. Also, saving things directly from the Media tab on the Firefox Page Info window still seems to work for the most part with the current page coding, since Instagram seems to be surfacing the better versions directly on the pages now, it's just not as convenient.