I had the same problem, and I read this thread, then I was able to fix it.
I'm not sure which of these changes to the amped wireless repeater made the difference. Now that I have it working, I'm not going to make individual changes to see what fixed it. I bolded the changes I think actually made a difference below, so you can all do them one at a time and see what fixes it.:
Under Wireless Settings --> Security:
Changed wireless security to WPA-Mixed, personal, AES, Passphrase (this was actually my original setting, but I had changed it back and forth to try to fix the problem. I don't think the security protocol was the problem, but I include it only because it is part of the final configuration that actually worked)
Under Wireless Settings --> Extended Network
I added a guest network. I don't think this made any difference. I was adding a guest network so that I could try connecting chromecast without any encryption - but I got it to work before I had to try that.
Enabled WMM. I think this is what made the difference. It had originally been set to "disabled." Now that I have it enabled, the option is dimmed out, and I can't change it anymore. I'm not sure what setting elsewhere in the router I changed that temporarily allowed me to change this setting, but I really think this setting is the one that made the difference. It is a patch to allow 802.11e functionality. I'm no expert, but I believe 802.11e has to do with secure device recognition. I guess they were taking a long time to finalize the standard, so a lot of industry types created this temporary cludge called WMM. Anyway, it should be enabled.
Under Wireless Setting --> Advanced settings
Disabled "Protection" This was already disabled when chromecast wasn't connecting, but I include it so you know it wasn't the problem in my case. I can't figure out what this setting actually does, but it may be a problem for some people.
Disabled WLAN Partition: Again, this was already disabled by default on my network, so it wasn't causing my problem. However, it may cause some people problems.
Enabled 20/40MHz Coexist. (I was originally only broadcasting with a 20MHz channel width. I don't know which setting here is correct, but coexist is enabled in my final config)
Under Home Network
Channel Width 40MHz. This may be important, because you really want the 40 setting for better speed for video casting. Chromecast may be set up only to work with 40MHz... I don't know.
Control Sideband: Upper
Under IP Settings
I disabled DHCP. This seems to work best for my network. Everything gets messed up when you have two DHCP servers on the same network (the cable modem/router AND the Amped Wireless range extender. When I enable DHCP, I get all kinds of network instabilities. I set the IP at the top to 192.168.1.4. The main cable modem passes through the range extender and handles all DHCP IP address assignments very well.
Default Gateway 255.255.255.0 (I didn't change this)
Default Gateway 0.0.0.0 (I didn't change this)
Domain Name: I deleted it and left it blank. This didn't fix the problem, so I don't think it matters.
802.1d spanning type: Disabled. When I enabled this, my whole network got unstable. IP address conflicts etc. It's weird, because I thought this protocol was supposed to fix those problems. Anyway, "disabled" was the default, and it was set both ways without fixing the chromecast connection. Do what you want with it.
UPDATE:
I now know which fixed it...
Chromecast doesn't work with WEP encryption. It needs WPA.
To explain: After I got everything working, I decided to upgrade the firmware. Then it wouldn't reload my settings properly from the backup file, so I had to manually go in and reconfigure everything. The new firmware had a cool option to clone network settings from the main network (My cable modem/router), so I tried it and set everything else as I mentioned above. It didn't work. The only difference was the bad configuration was now using WEP encryption instead of WPA. So, I switched back to WPA-Mixed (as in original post), and it worked again.
Hope that helps, everyone.