user1622: I am trying to find some sort of wifi adapter for Arduino, it needs to be as small as possible (not a shield) and as cheap as possible. 
So far I can only seem to find shields that cost about £10+ but I figured that if you can get a replacement wifi adapter for a laptop for next to nothing that I should be able to get a wifi adapter for Arduino much cheaper than this.
It's possible I just don't know what to search for.

user27: The Roving Networks RN-XV WiFly module (found sold at €30) might be a solution. I've never tried it. It supports 802.11b/g.

user70109: https://www.forward.com.au/pfod/CheapWifiShield/index.html
provides a very inexpensive wifi addon for Arduino

The WiFi Shield needs just two parts, 4 bits of wire and some soldering equipment. Parts List:-
Adafruit HUZZAH ESP8266 Breakout US$9.95 + shipping
Uno Protoshield US$1.88 + shipping
Total US$11.83 (as of June 2015)
For an even cheaper version ~US7 see https://www.forward.com.au/pfod/CheapWifiShield/ESP2866_01_WiFi_Shield/index.html
Both of these need a US10 USB to TTL cable to program them
After you have installed the pfodESP8266BufferedClient library, open the Arduino IDE and copy this sketch, ESP8266_WifiShield.ino, into the IDE.
That provides this setup webpage.


user653: Some months have passed since my previous answer and now the ESP8266 based WiFi modules are available.
They get connected to your project serially (3.3V) and are controlled in a modem like way using AT commands.
With a pricing of round about $5 they fit the question better than my previous OpenWrt based answer.
For more information about ESP8266 ask your favorite finding machine or list news with tag ESP8266 at Hackaday.

(Old answer from the pre-ESP8266-days:)
I vote for a small router reflashed with OpenWrt like "TP-Link TL-WR703N".
It is like building an own Yùn with an Arduino of your choice...
The WR703N:
It has Ethernet, 150MBit WiFi, USB and if you want to solder, a serial console and 2 GPIO-Pins which can be turned into I2C.
This sure is not a chip- or stamp-sized solution but often cheaper than the alternatives, more powerfull and extendable.
Search the net with your favourite find'o'bot for "arduino wr703n" and you'll find lots of examples...

1622: <p>I am trying to find some sort of wifi adapter for Arduino, it needs to be as small as possible (not a shield) and as cheap as possible. </p>

<p>So far I can only seem to find shields that cost about £10+ but I figured that if you can get a replacement wifi adapter for a laptop for next to nothing that I should be able to get a wifi adapter for Arduino much cheaper than this.</p>

<p>It's possible I just don't know what to search for.</p>
 27: <p>The Roving Networks RN-XV WiFly module (<a href="http://www.watterott.com/de/Roving-Networks-RN-XV-WiFly-Module" rel="nofollow">found sold at €30</a>) might be a solution. I've never tried it. It supports 802.11b/g.</p>
 70109: <p><a href="https://www.forward.com.au/pfod/CheapWifiShield/index.html" rel="nofollow noreferrer">https://www.forward.com.au/pfod/CheapWifiShield/index.html</a>
provides a very inexpensive wifi addon for Arduino</p>
<p><a href="https://i.stack.imgur.com/OjJAA.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/OjJAA.jpg" alt="Completed Shield" /></a></p>
<p>The WiFi Shield needs just two parts, 4 bits of wire and some soldering equipment. Parts List:-</p>
<p>Adafruit HUZZAH ESP8266 Breakout US$9.95 + shipping<br />
Uno Protoshield US$1.88 + shipping<br />
Total US$11.83 (as of June 2015)<br />
For an even cheaper version ~US7 see <a href="https://www.forward.com.au/pfod/CheapWifiShield/ESP2866_01_WiFi_Shield/index.html" rel="nofollow noreferrer">https://www.forward.com.au/pfod/CheapWifiShield/ESP2866_01_WiFi_Shield/index.html</a><br />
Both of these need a US10 USB to TTL cable to program them</p>
<p>After you have installed the <a href="https://www.forward.com.au/pfod/pfodParserLibraries/pfodESP8266BufferedClient.zip" rel="nofollow noreferrer">pfodESP8266BufferedClient library</a>, open the Arduino IDE and copy this sketch, <a href="https://www.forward.com.au/pfod/CheapWifiShield/ESP8266_WifiShield.ino.txt" rel="nofollow noreferrer">ESP8266_WifiShield.ino</a>, into the IDE.
That provides this setup webpage.
<a href="https://i.stack.imgur.com/nYvAK.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/nYvAK.png" alt="Config webpage" /></a></p>
 653: <p>Some months have passed since my previous answer and now the ESP8266 based WiFi modules are available.</p>

<p>They get connected to your project serially (3.3V) and are controlled in a modem like way using AT commands.</p>

<p>With a pricing of round about $5 they fit the question better than my previous OpenWrt based answer.</p>

<p>For more information about ESP8266 ask your favorite finding machine or <a href="http://hackaday.com/tag/esp8266/">list news with tag ESP8266 at Hackaday</a>.</p>

<hr>

<p><em>(Old answer from the pre-ESP8266-days:)</em></p>

<p>I vote for a small router reflashed with OpenWrt like "TP-Link TL-WR703N".</p>

<p>It is like building an own Yùn with an Arduino of your choice...</p>

<p>The WR703N:</p>

<p>It has Ethernet, 150MBit WiFi, USB and if you want to solder, a serial console and 2 GPIO-Pins which can be turned into I2C.</p>

<p>This sure is not a chip- or stamp-sized solution but often cheaper than the alternatives, more powerfull and extendable.</p>

<p>Search the net with your favourite find'o'bot for "arduino wr703n" and you'll find lots of examples...</p>
