LessWrong developer here. Here's an overview of what all those domains are. The code is open source, so you should be able to verify these, with some effort.
Algolia (algolia.net, algolianet.com) is a service we use for site search (what you get when you click the magnifying glass icon on the top-bar). They have a mirror of all searchable data (ie non-draft posts and comments, tag pages, user bios); they receive a copy of searches that are performed through the site search box, which they can associate with IP addresses but not with usernames.
Cloudinary is an image-hosting CDN that we use for images in some posts and images that are part of the site UI.
dropbox.com and dropboxusercontent.com are hosting images that were used in posts, presumably because they were visible in the Recent Discussion section when you loaded the front page. Currently, when users insert images into posts, depending how they do it and which editor they're using, it may point to the original domain of the image. Also, for authors we have set up automatic crossposting for, the crossposts will use the original image URLs. We will hopefully switch this to always upload those images to Cloudinary and host them there instead, partially for privacy reasons but mostly to prevent link rot in archives of old posts.
dl.drop is not a valid domain name; it's either a broken image link in some post that was in Recent Discussion, or a typo in this post.
The Google domains are from Google Analytics, Google Tag Manager, Google Fonts, and ReCaptcha. Google Analytics and Google Tag Manager measure site traffic and aggregate usage patterns.
intercom.io is for the chat icon in the bottom-right corner, used for messaging the admins about the site.
lr-ingest.io is LogRocket. We (the devs) use it to see how the site is being used; we can watch anonymized replays of sessions (anonymized in that the username in the corner is edited out). As policy, we don't read people's direct messages or unpublished drafts, or deanonymize votes, though in principle we have the capability to (both with this tool or with direct database access).
TypeKit, aka Adobe Fonts, is a font library and font hosting service. We could probably consolidate this with one of the other CDNs being used, but font-hosting involves some user-agent-string based compatibility polyfills, which would be somewhat annoying to reproduce ourselves.