I can't understand for the life of me what's going on anymore, my project was working fine at home, but now I'm away from home and it developed an unpredictable behaviour.

I fixed the ESP restarting troubles somehow, but now it refuses to connect to the access point. I'm using the SDWebServer as the base example, which works great without modifications, no issues whatsoever. This is the serial debug output:

Connecting to 4a82dd
scandone
state: 0 -> 2 (b0)
state: 2 -> 3 (0)
state: 3 -> 5 (10)
add 0
aid 3
cnt

connected with 4a82dd, channel 11
dhcp client start...
chg_B1:-40
chg_B1:-80
ip:192.168.0.20,mask:255.255.255.0,gw:192.168.0.1
Connected! IP address: 192.168.0.20
MDNS responder started
You can now connect to http://esp8266sd.local
HTTP server started
SD Card initialized.
pm open,type:2 0

I added OLED screen support (uses Wire library) and some other bits and pieces and now it simply cannot connect to the AP. Sure enough by commenting it out the Wire library calls the thing works as expected again... but I don't have a screen anymore, no good! This is the output:

Connecting to 4a82dd
scandone
state: 0 -> 2 (b0)
state: 2 -> 3 (0)
state: 3 -> 5 (10)
add 0
aid 3
cnt
chg_B1:-40

connected with 4a82dd, channel 11
dhcp client start...
chg_B1:-80
Could not connect to4a82dd
pm open,type:2 0

It's like the Wire library interferes with the WiFi somehow. I need to test this on another ESP because I'm about to accuse the hardware at this point.