Even if medical expenses aren't the kind of catastrophic, urgent expense you might face, there are other kinds of emergencies you might face - which might vary depending on your personal circumstances. For example:
When I was in my 20s and 30s (single, paying stabilized rent, stable job in Canada that I commuted to on public transit), I didn't have to worry about high medical expenses, a massive jump in my rent, sudden housing expenses, a dead car etc. Probably my biggest unpredictable expenses were veterinary bills for my cat.
In my 40s (married, living in Europe, paying stabilized rent, stable job that I commuted to on public transit), most things were the same but I did have to pay suddenly for several trips home (tickets for 2) to Canada when my mother became ill with cancer and then died. Thankfully, we had a very good leave policy and generous bereavement leave, so I didn't have to take any unpaid leave days. But I still had to come up with about $10K one year that wasn't in my regular budget. And I exhausted my leave that year so if I'd needed to take more time (and I was the sole income earner), I would have needed another $5K per month.
In the dumpster fire that was my 2022, we had to deal with, among other things, multiple cross-country trips to help care for my elderly father and stepmother (both had cancer, one died), a dental emergency that was partly covered by insurance (but still required an outlay of about $1K), a surprise $45K tax bill in a down equity market (because my Vanguard funds paid a whopping $115K deemed dividend that was historically anomalous), and then a requirement to pay quarterly tax instalments (of $45K) for anticipated taxes in 2023. Add to that, we got a free car from my father in 2023 (yay) but had to pay to get it moved across the country ($1500) and also received a beautiful, very large painting from him (inherited from my grandfather and probably worth about $15K) but also had to pay $750 to move it across the country. Husband also had to deal with some tax problems that required an upfront payment ($20K) that will likely get reimbursed when the issue is resolved (relating to proof of payment of US taxes). We also moved my father into a retirement home - thankfully, he does not need any financial support. Also thankfully, our relatively young house (about 7 years old) doesn't have any major structural issues.
We did have an emergency fund, and a line of credit, but it didn't stretch to the amounts needed (and needed suddenly) for all those 2022 bills - and in several instances we needed to come up with thousands, or even tens of thousands of dollars in days. We also had financial assets that could be liquidated, but the market was dropping so it wasn't a great time to do that (and interest rates were going up on our LOC).
A few lessons learned from 2022: With aging parents, there is a need for a bigger cash cushion. For one parent, all we really need to think about is the cost of sudden travel and accommodations. For the other surviving parent, we may need to suddenly start paying for part of their nursing home costs because that parent has very limited resources and provincial funding for nursing home care is only for the basics (and there are waiting lists for subsidized care in good nursing homes). We also need to be prepared for weird tax bills because of our financially complex situation. And as our three cats age (they're 10 now), we may be facing some large, sudden vet bills in the coming years. We also want to have resources to deal with house issues, especially as weird weather increases the risks of things like storm damage.