My business is... troublesome.
What I do is legal in every country on earth, but some people don't like it, and make it so tough on my poor ISPs that I am forced to go looking for new providers more often than I would like.
The only option I know of for somebody in my position is called "bulletproof hosting" (think wikileaks) and it isn't cheap.
The only thing that makes bulletproof hosting expensive, or anything different than typical hosting, is their legal stance when responding to "abuse" complaints via email. Your typical host will get fed-up after 5 or 10 no matter what the reason, and a bulletproof host will take the time to look through the legality of the matter and make a decision based on that.
As far as I know, these abuse emails are directly tied to the ip address on which their server sits, because they "own" that ip address and have the ability to lease it out to me on my expensive bulletproof server.
If I could answer them personally, I would save another company the trouble and hopefully save myself some money along the way.
How can I become the bulletproof host? Just rent out a room in my local DC and ask where to get the IPs from?
P.S. No... just because I know this will be the first question everyone asks - I am not some spammer or rule 34 pornographer, what I do is legal in every country. I said "think wikileaks"!
EDIT: Thank you once again for all of the amazing responses. Don't know why I've been lighting fires here recently. Thanks to everyone who saw through the smoke and provided me with meaningful answers.
You need to apply and be granted your own IP allocation by your local registry like RIPE or APNIC.
They require annual fees, and you need to justify your requirement (yours is legit). They will assign you an Autonomous System number and a range of IP addresses.
You must then find people to peer with (in a datacenter usually), preferably more than one. You then publish your BGP routes using your AS provided with your IPs via your new peer links.
You also should allocate a DNS server to provide PTR records for your allocation.
None of it is cheap, it requires expensive subscriptions, expensive peer links, expensive hardware (routers to do the BGP peering) and a fair amount of networking knowledge.
What I suggest is that you hire a contractor to set up the initial network and peering and firewall, etc to take the pressure off you having to learn and maintain it all, so you can concentrate on... Whatever it is that you host.
Who said that "bulletproof hosting" is expensive?
PRQ, the company that hosts thepiratebay.org, offers dedicated servers for less than $200 a month and simple web hosting for $10 a month.
From their website:
According to the wikipedia page they're hosting websites that promote paedophilia, so it should be no problem for them to host whatever you need hosted.
EDIT according to the same wikipedia page, PRQ also hosts wikileaks itself.
The "bulletproof hosting" does not work because of the ISPs reading and evaluating every abuse complaint, but simply because they are throwing every abuse complaint into the bin. This is certainly not expensive and certainly cannot be performed more efficiently if you do it yourself.
Apart from that the internet looks quite hierarchic if you do not happen to be a tier-1 carrier. There is always a possibility that complaints go to your upstream providers where they will be processed and your "bulletproof hosting" taken down - see the history of McColo for example.
If you need resiliency, you would need to spread the risk and use several providers with mirrors of your services across the world.
"If I could answer them personally, I would save another company the trouble and hopefully save myself some money along the way."
Have you provided a place on your site for users to submit feedback? This will prevent many users from going directly to your ISPs with complaints and allow you to state your case to them before they take further action.
Have you set up a FAQ page with proof that what you are doing is legal? This is essential in your case because it will reduce the number of complaints you get, especially if you recognize the common complaints and update the FAQ accordingly. Also, if your ISPs do shut you down, you can point them to the FAQ, which might give them enough peace of mind to get you up and running again.
If you haven't gone this route yet, this is definitely the place to start, as it is relatively inexpensive. If you have, this answer is not for you :)
Two other issues to think about as well.
There are a few alternate DNS providers available (opennic for example), setting it up now and letting your audience know what the fallback name is can go a very long way for you, considering what just happened in the US with the DNS confiscations by ICE. Static IP addresses wont help you unless your core audience knows them by heart.
Physical security will always be problematic, but this isn't something that you can stop (unless you're willing to apply a "kinetic solution"). So build a good offsite backup system. Also consider encrypting your disks, but encryption in this case is not a solution (if you run afoul of a new law, governments are willing to lock you up until you divulge the keys, consider this).
Good luck with this, what ever it is ...
Your best bet will actually be to find a major provider willing to partner with you to secure your site. If you really do have a legal and legitimate site you should be able to find one. A company willing to stand up for your rights that has the strength to with stand the initial press of they association with you will probably be the only thing that will provide stability to your hosting needs.
Another low-cost hosting provider that doesn't get fed up with abuse complaints is Nearlyfreespeech.net.
https://www.nearlyfreespeech.net/help/abuse