I really appreciate the different approaches here to thinking about an emergency fund.
Historically, we always had a very small emergency fund (maybe a couple of months). At that time, we both had very stable jobs and a high savings rate, so it was easy to very quickly save up for something like a roof or HVAC or whatever, and we had no fear of losing our jobs. Last year, my wife quit her job and I changed mine, plus we moved from the US to Canada. We are also quite close to (if not at) our FI number. We currently have a much bigger emergency fund (~18 months). I am investing some of it, but we've kept it bigger for a couple of reasons. Number one, we weren't sure what was going to happen with our jobs. DW is starting to work part-time, and my job seems stable enough, so I think we can back off a bit. Also, we wanted to have a cash cushion in both currencies to avoid having to exchange when the rate was unfavorable. Starting next month, I'll be paid in Canadian dollars, so that will help there as well. Since we're closing in on FI, though, I still think we'll keep more cash than previously, just so we wouldn't have to sell equities when things are down.
For us, I think about it less as an emergency fund and more as part of our non-equity portion of our portfolio. We're about 85% stock, 15% bond and cash. As our cash position has grown, I've let our bond position shrink (by not rebalancing into it).