Raspberry Pi 2B... I have tried lots of cheap devices... If you want to go cheap the best bet is a raspberry Pi. Comparatively speaking the memory is limitless, you can install all you want... It needs a good USB antenna when can be pretty cheap, just make sure the chipset is supported. With a usb flash drive one can run transmission and download torrents without leaving a computer on... The Rpi3 is good too, but one would probably want to use an external antenna anyway, for better coverage... Ah, one problem is that there is only one ethernet port. I used a usb ethernet dongle for a second port, and put it on the WAN ADSL side. It manages to keep up with my ADSL download bandwidth...
I have a lot of "favorite cheap devices", depending on what I use them for. But if you want to really get your device cheap, my first and foremost recommendation would be:
Get your device second-hand.
Virtually all of the 20-or-so devices I have amassed here over the past years I bought second-hand, sometimes even described as "broken", or as cheap as one or two euros. And I have yet to encounter one that didn't work.
My favorite "cheap device" right now would have to be an Easybox 904 xdsl. LEDE support is just starting out right now, but it's already a really, really nice all-in-one box if you're living in Annex-B land. They can be had as cheap as €10 plus shipping, I bought mine for €1.
I’d like to mention the Blueendless BS-U25AWF. It’s basically a hard drive enclosure for a 2.5 inch drive with the MT7620 chip at 580MHz, 16MB of flash, 64MB of RAM (though my variant has 128MB for some reason), a USB port, a battery, and has dual stream (2.4ghz only) WiFi with an 100Mbps Ethernet port. It’s supported by the snapshot versions currently but works great for a little portable file server.
Used youkuYK-l1. It can be 5 $.
32MROM / 128MRAM / 8GBTF / MT7620a
There is currently openwrt tf card support issues, if solved, comes with 8GB TF card will be very fun. Use overlay it can hold more things.
I now add a U disk used to install octoprint. It can replace raspberry pie as a 3D printing host computer, providing wifi online printing service.
In fact I think you should not try any MT7620 solution for the router
2.4GWIFI of MT7620 scheme does not work normally from 2014 to present (for example, if you plan to use OPENWRT or LEDE, the router of MT7620 scheme does not work wirelessly)
Although the MT7620 is very cheap
I think ASUS RT-ACRH13 will be a better choice
LEDE works perfectly well on the ASUS RT-ACRH13 (there are even reports that the LEDE has a higher wireless rate than the official firmware's wireless rate)
They're pretty minimal OpenWRT devices with 8MB flash and 32MB RAM, but the plug-top form factor, ethernet, wifi and USB ports make them perfect for wifi enabling ethernet devices, or network-enabling USB peripherals of all varieties. Only one port carries the USB signal, the other is a high-current charging port, suitable for powering high drain USB peripherals that get power from two ports via a Y cable (eg hard drives). I have one running a printer, one running a hard drive, and one currently serving as a wifi repeater.