Q:

Alternatives to Overleaf (i.e. instant TeX compiling without sign in)

The best thing about Overleaf  V1 was that I could log into it without an account. This made it possible to share things with people who most likely wouldn't be big TEX users and to whom the idea of working in "code" is scary. 
Now having a login, any login, is a impediment for this. If I can show them to the result and TEX markup side by side then I have a chance, however small, for them to read it and see that it's not that fancy.
So my question is is there a workable alternative to the new Overleaf? A tool that has no login requirements, so that I can directly link to the source to updating result view, with edit support for casual users coming in the first time. And with no confusing login requirements, not even ones you can circumvent.

A:

Log date April 2019 (ever changing market)
Bigger players once they reach a certain point (success) do not want the headaches associated with random demo users so I will simply mention those currently failing into that category such as Overleaf, Authorea, CoCalc and Datazar. All seem to have a large enough freemium/paying customer base. 
There is a separate "Big List" which includes other online products such as Google Docx2LaTeX, Verbosus and similar browser plug-ins that either require installation or handing over personal details for basic entry.
This answer (A "Small" List) is confined to current online with free access editor/compilers without a dependency on any install. Equally to be listed here there must be no mandatory registration e.g. social account for basic entry (although logins are sometimes common for "added" features / packages).
It is the smaller freemium players that are left behind trying to get a better share of the market.
Currently only a few TeX Live without registration landing page editors are still active. My fist test is will they run multiple classes such as including Beamer, without sign in.
1) Main contender, Papeeria (good spellchecking, autocomplete editor with most packages available)
https://papeeria.com/ (TeX Live default 2015 but can opt to use 2016)
2) TutorialsPoint have good support for packages such as beamer but less editor features, note heavily biased towards user hitting adverts (CodingGround serving Tex Live 2016) https://www.tutorialspoint.com/online_latex_editor.php
3) uni-halle Very spartan interface but can produce beamer 10 image presentations etc.  https://latex.informatik.uni-halle.de/latex-online/latex.php 
4) Tex Viewer, Editor (2015+ Version Multiple UTF-8 language formats supports Beamer) https://texviewer.herokuapp.com/
5) TeXonWeb (Jan Prichystal's older 2011 Tex Live. Limited to basic (La)TeX formats but does include Beamer class) https://tex.mendelu.cz/en/
 
Also Ran
https://latexbase.com/ (Limited number of Main Packages e.g. `beamer.cls' not found however can work in offline mode
Troy Henderson's
http://www.tlhiv.org/ltxpreview/ (more recent limited range of 2017 packages) and MetaPost previewer http://www.tlhiv.org/mppreview/
There is a very up-to-date (LuaTeX, Version 1.10.0 (TeX Live 2019)
ConTeXt online compiler at https://live.contextgarden.net/
Hybrids (can switch from simplistic equation editor to Latex syntax) 
In minimal or full latex mode will require \documentclass{} but may be limited to PNG rather than PDF output 
https://www.latex4technics.com/  (older packages may fail, good for simple  PNG output)
http://sciencesoft.at/latex/?lang=en  (can store tex and pdf online for sharing)
http://sciencesoft.at/latex/index?ochem=true&lang=en (specialist OCHEM-LaTeX)
Smaller equation editors (some possibly should be included above ?)
CodeCogs Equation Editor https://www.codecogs.com/latex/eqneditor.php
Interactive LaTeX Editor https://arachnoid.com/latex/
Noteworthy (thanks to @sebastiano)
Conversion from Latex to MathML
https://www.mathtowebonline.com/
To be continued...

A:

I'm a co-founder at Overleaf. Thanks for your feedback.
I think the underlying question here is about Overleaf requiring users to log in in order to edit, which I can comment on.
We used to allow anonymous editing --- that is, if you sent collaborators the secret link for a project with link sharing turned on, they could edit without logging in. However, we found that this lead to a steady stream of difficult support requests and unhappy authors. People in a panic would demand to know the identity of the "anonymous user" who had just deleted or rewrote some important part of their thesis or paper, which we could not tell them. People would edit their own projects when they were not logged in and get confused, because their changes showed up as being from an anonymous user. In some cases there were allegations of foul play from wayward collaborators or account hacking, but in the cases I was involved in, it was always just a misunderstanding.
I know this was a useful feature for some users, and it was an important one for me personally when I built it into the original Overleaf. However, with the benefit of several years' experience, I came to view it as a "footgun". Overleaf now has a community of over 4 million authors, so a even a feature that is useful 99% of the time but problematic 1% of the time can represent a major drain on our resources. I therefore argued successfully for its removal as part of our Overleaf v2 project.
There were several other factors that supported this decision:

Anyone can register for an Overleaf account for free. We support email and password as well as one-click Single-Sign-On options, such as Google and ORCID.
We allow anonymous read only access for projects via the 'read only' link (when the owner has enabled link sharing).
Fundamentally, it is very useful to be able to see who has changed what, and this supports our main mission: to make it easier to collaborate on LaTeX projects.

We are considering ways that we might be able to safely bring back more features for anonymous users, for example to allow them to leave tracked changes or comments, either on the source or the PDF, without allowing them the full power of editing, which we know was problematic.
Finally, since open source has also been mentioned as a criterion, I should point out that Overleaf v2 is open source (https://github.com/sharelatex/sharelatex) following our integration with ShareLaTeX in 2017.

