Google Messages for Web (https://messages.google.com/web) script to save an entire conversation history in simple text form. MMS messages such as images are not saved.

google messages chat backup.js

// Chat conversation backup for https://messages.google.com/web
// To use, start by choosing a conversation, and scroll to the top until all previous chats have been loaded. 
// Then copy and paste this script into the Chrome Developer Console (Ctrl+Shift+J), hit Enter, and copy the 
// value of 'result' and paste it wherever you would like to save the chat history.

var result = Array.from(document.querySelectorAll("mws-relative-timestamp, mws-text-message-part")).map((item) => {
  switch (item.nodeName) {
    case "MWS-TEXT-MESSAGE-PART":
      // Extract the sender's name from aria-label, which starts as '[Person] said:'
      var aria = item.getAttribute("aria-label");
      var idx = aria.indexOf("said: ") + 6;
      return aria.substr(0, idx) + item.innerText;
    break;
    case "MWS-RELATIVE-TIMESTAMP":
      return item.innerText;
    break;
  }
});
result.join("\n\n");