This is probably the simpliest and cheapest WiFi chip we can find. It is arrived on the market at the end of 2014 and sold worldwide at beginning of 2015.
This module has 8 PINS too:
Pin 1
TX
Data out.
Pin 2
GND
Ground
Pin 3
CH_PD
Power down, pull up to leave enabled
Pin 4
GPIO 2
IO, not used
Pin 5
RESET
Pull up if you don't want any hard reset
Pin 6
GPIO 0
Input, pull up. If set to GND it is in programming mode
Pin 7
3.3V
3.3V to 3.6V (not 5V)
Pin 8
RX
Data in, can be read as serial with 115200 bauds
It has 96kb of RAM and a lot of IOs, so probably I could write the entire code directly on this chip... But there is no or poor documentation, so the Arduino is not bad for it, also if we have only 2kb. What we need is RX and TX for the AT-commands and 3.3V (no it doesn't work with 5V!). There are a lot of sellers, so probably not every ESP8266 ESP-01 is the same... Some modules become really hot, other have a wrong resistor, others will destroy the firmware if you send the wrong AT commands, and so on...
Schematics
There are different ways to link it to the arduino, this one is the easiest. A better way would be with a logic level converter. Remember to pull up CH_PD, RESET and GPIO0. There are a lot of firmware out there. Maybe flashing to a stable one is a good way if the module doesn't work anymore! This firmware v0.9.5.2 AT Firmware.bin from www.electrodragon.com seems to be working very well and you can also read the GPIO2 with the AT+CIOREAD=2 command.