The following is a list of browsers that I test on.
Chromium (latest): This one is obvious. Half of Internet users use Chrome, so having some variant available is very useful.
Firefox ESR: This is the Firefox (Gecko) version most readily available to me since I use Debian.
Epiphany/GNOME Web: I use this as a representative of Webkit, which is used in Safari and many other browsers.
Microsoft Edge: Uses a unique web engine and thus necessitates testing on it. Since I use Linux I don't usually have this available.
Palemoon: Uses a fork of Gecko called Goanna, thus making it differ from Firefox in some ways.
Chrome on Android: The default browser for Android, used to test mobile. This is my primary browser when testing on mobile devices,
Firefox Focus: Blocks external sources, demonstrating how websites look when external sources are blocked. Also shows Gecko on mobile.
Safari on iOS: The default iOS browser. Used to test mobile on iThings.
Current browsers I don't test on:
Internet Explorer: Internet Explorer has been in LTS for years, very few people use it, and it's lagging behind is very well-known, so I take test results there with a grain of salt (as should you).
NetSurf: Netsurf is designed to be lightweight. It lacks many modern standards, and thus won't render well.
Links: Also designed to be lightweight and thus only supports partial HTML4.
Browsh: This text-based browser is basically a pixelated Firefox, and is the same except for appearance differences.
Anything not listed: Browsers like Vivaldi don't require testing because they use the same web engine as another browser.
Old-style text-based browsers like w3m, EWW, and Emacs/W3: Only one text-based browser needs to work, since others implement similar functionality. All that really matters is that the document is organized.
Basically, I want to test supported versions of the Blink, Gecko, WebKit, EdgeHTML, and Goanna on desktop, tablet, and mobile devices. Responsive mode is rarely enough to test this, since responsive mode doesn't emulate the quirks that mobile devices have.