I found and old WRT3600 in the basement and I think I once installed OpenWRT/Lede on it. I cannot connect to it anymore, so I go into failsafe mode. According to the flashing light, it seems to work. However, I cannot find the router at 192.168.1.1. An nmap scan of the entire 192.168.1.0/24 range yields nothing. Same with 192.168.0.0/24.
Any idea what could be done?
nmap 192.168.0.0/16 
Alternately, try 192.168.7.0/24 (I have a vague recollection of that subnet being used at some point.
Another option might be to run tcpdump / Wireshark on a device connected directly to the router. See if the router spits out anything recognisable by the listener.
nmap 192.168.0.0/16
I did that, but if found nothing, so that excludes 192.168.7.0/24 too.
tcpdump / Wireshark
I have no experience with those, so I'm probably out at this point.
Did you also give your computer's network interface a /16 subnet mask before scanning? If you try to scan the /16 address space from a client in a /24 address space, it'll try to send most of the scanning traffic via a router, which may not be helpful here.
Good opportunity to learn. You'll be trying to identify if there's anything coming from a device which isn't currently being used, so there's little risk of breaking anything in the process.
could also try assigning an IP to it (in your subnet) on your computer using arp.
Did you also give your computer's network interface a /16 subnet mask before scanning?
Yeah, I did that. No luck.
Good opportunity to learn. You'll be trying to identify if there's anything coming from a device which isn't currently being used, so there's little risk of breaking anything in the process.
Alright, I just looked at tcpdump's output, and the only suspicious thing I could find is
14:58:14.232986 IP 10.230.47.194.698 > 255.255.255.255.698: OLSRv4, seq 0x4415, length 20
I can't ping 10.230.47.194 though. Any idea what to with this?
Just had a thought: what happens if you set your computer's network interface to DHCP, and then connect it to each available port on the router in turn? Does your computer pick up an IP address from any of the available ports? Also, if you have tcpdump listening in the background, does it pick up any of the DHCP DORA sequence at any point?
Another thought: if you don't need to preserve any existing configuration, what happens if you perform a paperclip reset of the router?
Just had a thought: what happens if you set your computer's network interface to DHCP, and then connect it to each available port on the router in turn?
No DHCP packets anywhere to be seen. Just these wird OLSRv4 ones. (If not in failsafe mode.)
Another thought: if you don't need to preserve any existing configuration, what happens if you perform a paperclip reset of the router?
I tried that first of course. Nothing happens upon pressing the reset button for long. I only get into failsafe mode when pressing it at startup.
Is this the router in question? - https://openwrt.org/toh/tp-link/tl-wdr3600
If it is, does the "TFTP auto recovery" information help?
@frollic Thanks for the suggestion to use arp. Unfortunately,
arp -s 192.168.1.1 E8-94-F6-BB-AE-22
gives
arp: invalid hardware address
Try E8:94:F6:BB:AE:22 (colons instead of hyphens).
Is this the router in question? - https://openwrt.org/toh/tp-link/tl-wdr3600
If it is, does the "TFTP auto recovery" information help?
Yes. I tried following the instructions and it's all going according to what is described, but I can't start the atftpd server. This
atftpd --no-fork --daemon .
gives
atftpd: can't bind port :69/udp
Usual reasons for that are either insufficient privileges, or something else already has that port open.
For the first possible cause, are you trying to run atftpd as root/sudo ?
For the second possible cause, what's the output of either sudo netstat -alnp4 | grep :69 or sudo lsof -ni :69 ?
Try E8:94:F6:BB:AE:22 (colons instead of hyphens).
This does something, but I still can't ping or ssh into the router.
sudo netstat -alnp4 | grep :69
udp 0 0 0.0.0.0:69 0.0.0.0:* 274247/inetutils-in
sudo lsof -ni :69
COMMAND PID USER FD TYPE DEVICE SIZE/OFF NODE NAME
inetutils 274247 root 4u IPv4 4737870 0t0 UDP *:tftp
So it's the latter reason: something else already has that port open. A quick google of "inetutils" "tftp" suggests you might already have tftpd installed. If that's the case, you might not necessarily need to use atftpd. Alternately, if it is necessary to use atftp, you'll need to find a way to stop the existing daemon which is bound to that port.
Got it, killed the offending process. Let me try again...
It won't. The process name is inetutils. This page - https://www.gnu.org/software/inetutils/manual/html_node/tftpd-invocation.html - might be useful.
I finally got it! Flashed fresh OpenWrt and I'm ready to do. Thanks for all the help! 
What was the magic incantation in the end?