It's not going to be possible to control from the connection settings directly on your own computer. If you think about it you need control of each computer in the chain.
E.g. if you are talking about a chain of proxies you can only effectively specify one proxy (the next one along) either for the OS as a whole or a specific application (which can use a different proxy).
You would need access to the network settings of the computer hosting the next proxy in the chain which would no longer be on your local network (otherwise there would be no point).
So for example if you lease a virtual server on the internet, running SSH you can use that as a SOCKS proxy, but you would need to change the network settings on that server to use another proxy in turn.
Similarly you could in theory setup a chain of VPN connections if you control each VPN server or VPN gateway in the chain.
The IP you were browsing from would generally appear to be the public IP of the last (remote) computer in the chain away from you. There is no built-in setting on your own machine which changes how that last computer connects so you would have to remote access into it.
With some extra work (say a cron job) you could script your leased server to switch randomly between public proxies (if you can find a list of reliable ones) or public VPN relays (if you can find any non dodgy ones).
Otherwise you can combine using a SOCKS proxy and visiting an anonymous surfing website as you said.
However it seems like alot of trouble for a theoretical privacy benefit.
The simpler options would be things like:
1. the private browsing modes in modern browers
2. using a liveCD ISO run as a virtual machine (e.g. in VirtualBox).
There are a few cut down distributions or VM appliances with just a browser or browser plus simple apps that could run from a ISO with no data persisting between sessions,
For example download a Puppy ISO and create a VM in virtualbox using the ISO file as its CD.
Boot up Puppy and get it online.
When the browser is online take a snapshot.
You can restore to this snapshot at anytime with no browsing history persisting locally, leaving only sites logging the IP your are visiting from, which may be dynamic.
3. How about surfing from a PAYG 3G dongle. Some of these assign you a private IP behind NAT plus use a proxy server to save bandwidth by re-compressing images so unless the mobile operator requires registration details and keeps DHCP logs, web sites are only getting the public IP of the mobile provider's proxy server,
4. Very similarly on a mobile phone you could use Opera Mini as pages are delivered their proxy which reprocesses pages to a mobile-sized version.
But to echo another comment TOR seems to be designed to be closest to what you are trying to do.
prompt $P - Invalid drive specification - Abort, Retry, Fail? $G
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