Two very different problems, depending on whether the tab is still open.
Saving a temporary chat splits into two cases: the tab is still open, or it isn't. If it's still open you have several options. If you already closed it, it comes down to whether anything on your own machine was keeping a copy.
Skip to whichever one you're in.
ChatGPT and Gemini both call this a Temporary Chat. Claude calls the same thing an Incognito chat — the ghost icon in the chat header. Everything below applies to all three.
Free options, easiest first.
Ctrl + P, then set the destination to “Save as PDF”. Nothing to install. Long conversations sometimes get cut off and the layout can break — code blocks suffer the most.
Ctrl + A, Ctrl + C, paste into a text editor. Reliable, but you lose headings and code formatting.
Search the Chrome Web Store for “ChatGPT export” (or your own service) and you'll find several that write PDF or Markdown. Most are free and they're fine for ordinary chats.
One caveat: check that the one you pick works on a temporary chat. Many of them read your conversation history, and a temporary chat is never written to that history, so it can fall outside what they handle. Test it once before you rely on it.
The short version: you can't get it back from the service.
A temporary chat never enters your history, and once you close it there is no way to open it again. Signing back in won't help — there is nothing there to find. This isn't like undeleting a file. The same is true on all three services.
Which leaves one place a copy could exist: your own machine, if something was already keeping one.
That's the gap I built AI Session Archiver (ASA) to cover: a Windows app plus a Chrome extension.
While capture is running, every message is appended to a file on your own PC as it arrives, and ASA shows a capture window counting what it has taken.
Starting part-way through a conversation is fine. Scroll back to the top of the chat and everything said before that point is collected too.
Being straight about the limits:
So if you had both switched on for the chat you lost, open ASA before you give up. If you didn't, this insures you against the next time rather than recovering the one you just lost. Sorry.
The free edition is enough to try the saving itself — no time limit, no cap on how many conversations you save.
Setup_..._ASA.vbs insidePress it on any conversation you want to keep, temporary and incognito chats included. That much is free.
Steps 1 to 4 are the free edition. Step 5 needs the paid one. The extension is the same either way, so switching means replacing the app only.
Auto-recovery is one part of ASA. See “What ASA does” below for the rest.
Besides auto-recovery, the paid edition adds Markdown, plain text and PDF output, per-chat-type folders, and 50 filename tags instead of 3. $6 per edition or $14 for all three, one-time — there's no subscription.