Interesting ones include the NEXX 3020H/F/AD (8 MB Flash / 64 MB DDR / MT7620N / ~15 USD), the A5-v11 (4 MB Flash / 32 MB RAM / RT5350F / ~7-9 USD), Onion Omega2 (16/32 MB Flash / 64/128 MB RAM / MT7688 / 5/9 USD + ~5 USD shipping).
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Forget about a 4/32 device. The firmware size is slowly growing and 4 MB of flash will not be enough soon. You need something with at least 8 MB flash / 64 MB RAM.
https://lede-project.org/docs/guide-quick-start/start#why_we_require_8_mbytes_of_flash
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I know that. I actually own one, and my plan for it is to use the image builder to create a tiny image, then set up extroot with a swap partition. If I have to keep it on an old LEDE version, then so be it. I'm not using it as a router, anyway.
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These are cheap, but if you are willing to spend a few dollars more you can get a 16\64MB device
https://www.gl-inet.com/products/gl-mt300n/ $19.99 plus shipping
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I'd second MT300N. Also, the D-Link DIR-860L B1 I've owned briefly was inexpensive and pretty good for how much it costed.
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Nice. Very similar to the NEXX 3020H/F/AD. Depends on where you are and which are easier to source, I guess.
EDIT: The GL-MT300N also has twice the Flash memory. That's really quite nice.
EDIT 2: Sadly, it also seems to be about twice the cost (with shipping, from GearBest).
EDIT 3: Seems to be slightly cheaper than twice the cost from Amazon.
The DIR-860L is nice, but I think it's in a different price category. It's the type of thing I'd get to use as an actual SOHO router, but not as a component for prototyping/testing various projects.
Isn't $29,99 with VAT in local store for 16/128MB device with 2x2.0 USB not realy cheap? I have already pick up one for testing and porting.
Look at https://lede-project.org/toh/hwdata/tp-link/tp-link_tl-wr942n_v1
It is already on OpenWrt clone right out of the box, so porting LEDE is not a big effort.
I like TP-Link WR1043ND v4
It has QCA9563 750Mhz, 64MB RAM and 16MB Flash, 2.4Ghz WLAN and 1xUSB 2.0 port.
To be honest WR1043ND line has been pretty great from the beginning.
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I'm using TP-Link Archer C50 and I love it. 1 vote for Archer C50 
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Orange Pi Zero, Allwinner H2+, MicroSD, 256MB 7$, 512MB 9$, with case and expansion board 13$
https://www.aliexpress.com/store/group/Orange-PI-Zero/1553371_510589665.html
Ok, installation is a little unusual!
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I discovered the ZBTs last year if you really want to go cheap.
Their MediaTek based devices goes for around $9-12. They feature a web-failsafe mode which makes is a snap to flash.
https://www.alibaba.com/product-detail/ZBT-1626-openwrt-Wifi-router-OEM_60458250130.html
They also have a nice TP-Link 841 replacement which goes for around $16.
This one features 16M Flash and 128M Ram with USB Port and are included in the LEDE menu. (ZBT WE1526)
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Will they sell in small quantities? This seems to be an OEM site. Looks like a great option for mesh nodes.
You can probably find them in smaller quantities on AliExpress.
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Thanks. I found them. Unfortunately at 400% the OEM price
I don't have a real favourite yet. With cheap hardware there is always the question if certain hardware has all the options that you need
I like the GLI GL-150 beacause it has RJ45 LAN and WAN, and external antenna.
Still the DIR-505 is cheaper and has a PSU all in one package, but only one RJ45 and no external antenna. Still it's compact design makes this an interesting little router. Plug it into the wall and route.
If this would have had 2 RJ45's, 2 radio's and external antenna's then this could have been my favourite.
Then there is Orange Pi Zero. Not supported yet (or is it?) This is not really router hardware. But still looks interesting. It's cheaper, it has more CPU power and RAM. But also has a few disadvantages.
GLi's little routers offer a lot of choice for a reasonable price (not the absolute cheapest, but a fair price)
You might be able to contact the AliBaba vendor to buy a sample.
I wish it would have 1Gb ports instead of 100Mb/s
Only one ethernet port?
How is it even usable with OpenWRT?
I'd say try the onion omega kit.
1x Omega
1x Expansion Dock
1x Ethernet Expansion
This is the way to go for about 40-45 $
Still trying to find out if omega 2+ is capable of 1Gbps
@r43k3n Just read the second link in my post. 
If you build an AP or a wireless network bridge / CPE one eth is fine. And there is a lot of supported usb net stuff available in the market.