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I understand that IANA delegates public IP addresses to five Regional Internet Registries in which ISPs can purchase IPs from.
What is to stop someone not associated with IANA or any RIR from obtaining and using an IP address themselves (that may or may not have been delegated already)?
You are about 49% right. There is not much preventing you from using an arbitrary IP address. This can be done like this:
ifconfig eth0 99.99.99.99 up
(this appears to be a not actually used, but public address owned by AT&T, so please don't try this at home).
The first problem is to connect your host to the Internet. In my home my router provides 192.168.178.1 as the gateway to the Internet (the actual address the router of your ISP may vary). You have to send your packets to this address, since there's no other way to the Internet. Unfortunately, this gateway only accepts packets from 192.168.178/24, which 99.99.99.99 is not part of.
That's the first 1% of the problem you might face. With some manual routing table tweaking you may overcome this by persuading your network to send your packets to this gateway anyway.
Once you overcome this first obstacle, your packets will probably (with some luck) actually reach their final destination and will be processed there. A great share of layer 4 protocols in the Internet are TCP connections, though. They require the famous 3-way-handshake: So after the first SYN packet reached its destination the target host responds with a SYN|ACK package and sends it to the sender's address, which is 99.99.99.99.
Now your packet has to deal with the real 50% challenge: How to find its way home? Unfortunately you have told no one except yourself that you expect 99.99.99.99 to be delivered to your own host. And that is the main problem you have.
You may argue now, "Why can't I tell the Internet that I now have 99.99.99.99 on my computer?". Well actually, you could. You had to find a way for distributing this information to virtually the whole Internet (or at least all public Internet routers). Fortunately there is a means to do so. That's what we call BGP. It is run by a community of big boys that take care that a few rules are obeyed. This is more or less what we know as "buying and registering official IP addresses".
With some effort you could try and succeed to become one of the big boys and eventually become part of the community that runs the BGP infrastructure. If you managed to do so, you could actually tell the global BGP network to route 99.99.99.99 to your home. If you did so, you would have been expelled pretty quickly from that club, though, I dare to predict.
The short version of your question is it is easy to pretend you own a public IP address, but virtually nobody would take you seriously.
I'm not sure I know what you mean by obtaining an address outside of IANA as there are no means to do this, unless you mean to simply use a public address without it being allocated to you. If this is what you mean, then you can use any IP address on an internal network if you control the hardware making the routing decisions. Connecting to the public internet is another matter. ISPs filter BGP updates with their customers and only allow them to advertise address ranges that they know are allocated to them, so if you cannot advertise the address into BGP then traffic cannot route to you so, so the address would be useless.
Just think about it. When the IPv4 protocol was originally documented, it specified a 32 bit field for addresses. As soon as that spec was published (in September of 1981), there were 2^32 (about 4 billion) Internet addresses. The Internet community needed some way to arbitrate their use, so IANA was created (well actually Jon voluntered, and everyone else said "good, I don't want to do that").
How the space is used in a general way is set by the IETF (carving out the blocks for multicast, for example), but otherwise Jon (as the original IANA) started keeping a list and handing out addresses (originally Class A, B, and C later arbitrary prefixes). When the net grew and the job became too big for a single central authority, the regional registries came in as an intermediate level. And later, the job of doing the IANA record keeping was turned over to ICANN.
But that original set of addresses from 1981 is all there is or ever will be. That's why we are running out. The Internet was a much bigger success than anyone in 1981 could have imagined.
You can use any IP address you want. You most probably can't communicate with anyone outside your network.
To ensure proper communication on the internet addresses have to be unique. IANA is the organization in charge of address assignments.
