I'm now working for a 3rd party support organisation that has no contact with SAP, to the extent that I am not (based on independent legal advice to the company) allowed to access support.sap.com on any company device. I think I have avoided any reference to my employers name, so that this doesn't contravene any SCN guidelines 
If you cancel your SAP maintenance contract, you retain access to:
Security Notes
https://support.sap.com/securitynotes
Installation Data
https://support.sap.com/keys-systems-installations/installations.html
System Data
https://support.sap.com/keys-systems-installations/systems/view-edit.html
License Keys
https://support.sap.com/keys-systems-installations/keys.html
SSCR keys
https://support.sap.com/keys-systems-installations/keys/sscr.html
Migration keys
https://support.sap.com/keys-systems-installations/keys/migration.html
Of course, there is a distinction between going off maintenance because your release is out of date, and cancelling "official" SAP support entirely.
The former opens up a lot of business assumptions around core platform, risk assessment etc. I have worked with SAP customers who were on Max Attention or it's equivalent at the time, which made the support of 2.x and 3.x and 4.x R/3 systems relatively simple. We had more issues with hardware maintenance than software maintenance on those systems. Having said that, given the stability of those systems, you can see why customers who are happy with their only R3 system being a 4.x system don't see any point in having any connection at all with SAP.
In the case where you cancel SAP maintenance, you still own the rights to run and modify the software that you have rights to. What this means is if you are running ECC6 EHP4, you can download the EHP6 upgrade and appropriate OSS Notes, create the appropriate STACK XML for the upgrade, then cancel maintenance, then do your upgrade.
After you cancel maintenance, you retain your S number, but you lose the authorization to access notes or corrections (see exception noted below), documentation (except for SCN and help.sap.com), new software or patches, install the SAP router connection to SAP etc. This can cause problems; for example, sometimes the company I work for gets the job because the SAP system just isn't the focus of the business any more; not as important as most of us think it should be. This leads to, shall we say, an under-investment in training and staffing by the customer; I've worked with people who needed reminding that executing the next transaction in the SAP GUI needs a "/n" before it
Importantly you still have the rights to move your system (i.e. specify a new hardware key via the license keys tool) and to get SSCR Keys | SAP Support Portal to make changes to SAP objects and Z objects.
BTW, one exception regarding access to notes and corrections is that customers who have cancelled maintenance still have the right to access https://support.sap.com/securitynotes. I have to say it would have been easy for SAP to make this inaccessible, but it is genuinely good socially responsible behavior that should be applauded, as it must cost some effort (to distinguish between security and product issues) to maintain. The only caveat here is that SAP does not prvide operating system or DBMS security updates (except for HANA, of course).
What appears to be a more complicated issue is the customer who wants to run Windows 8.x and / or Office 365, but only has access to the SAP GUI 7.20 that was current when they cancelled maintenance. There are Microsoft technologies and techniques that help us get around that particular issue. More complicated examples exist as well, like another customer who had downloaded NW 7.4 software, but didn't create a STACK XML file. Theoretically, its possible they can upgrade, but we are legally / ethically unable to do things like suggest they use someone else's (on SAP maintenance) Solution Manager, I know it happens (we have many cases where SAP partners like IBM are providing first-level or hardware support to the Customer, and they will sometimes provide SAP IP that the customer isn't entitled to).
The worst case, from a technical perspective, will require that the customer "uncancels" SAP maintenance. They usually return (because of the issues mentioned elsewhere in this thread).
My final point is that I get paid based on case satisfaction surveys. I get a copy of the survey response, so does my boss, his boss, her boss, and the CEO. While the mechanics of our CRM system let me ignore or close a case any time I like, if I do so it gets reflected in the survey for that case, and in my pay packet. This breeds a close personal relationship between customers and Support Engineers; My customers have my phone number. Sometimes this gets abused, but you know what ? They may not always be right, but they will always be the customer
hth