WiFi repeater using ESP8266
You can't have a single ESP8266 be both an access point and a client, at least not using the exposed API (recall that the Arduino libaries do not implement much functionality themselves, all they do is call the underlying RTOS API).

Also, a signal repeater is not a radio-only device. It would have to understand the low-level 802.11* protocols and packet types so you don't end up with forwarding loops. You can't do it just by connecting a pair of 8266s back-to-back over SPI and blindly forward packets; there is a lot of networking code that would have to be written.