How can one tell the version of React running at runtime in the browser?

With the React Devtools installed you can run this from the browser console:

__REACT_DEVTOOLS_GLOBAL_HOOK__.renderers.forEach(r => console.log(`${r.rendererPackageName}: ${r.version}`))

Which outputs something like:

react-dom: 16.12.0

React.version is what you are looking for.

It is undocumented though (as far as I know) so it may not be a stable feature (i.e. though unlikely, it may disappear or change in future releases).

Example with React imported as a script

const REACT_VERSION = React.version;

let root = ReactDOM.createRoot(document.getElementById('root'));
root.render(
  <div>React version: {REACT_VERSION}</div>
);

<script src="https://cdnjs.cloudflare.com/ajax/libs/react/18.2.0/umd/react.production.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/react-dom/18.2.0/umd/react-dom.production.min.js"></script>

<div id="root"></div>

Example with React imported as a module

import { version } from 'react';

console.log(version);

Obviously, if you import React as a module, it won't be in the global scope. The above code is intended to be bundled with the rest of your app, e.g. using webpack. It will virtually never work if used in a browser's console (it is using bare imports).

This second approach is the recommended one. Most websites will use it. create-react-app does this (it's using webpack behind the scene). In this case, React is encapsulated and is generally not accessible at all outside the bundle (e.g. in a browser's console).

From the command line:

npm view react version
npm view react-native version