https://www.thenation.com/article/society/ssi-rules-families-poverty/
The Supplemental Security Income program (SSI) was created in 1972 under the Nixon administration to provide financial support to low-income seniors and disabled people. An effort to federalize state-level adult support programs across the country, SSI is a means-tested program—there are financial requirements to be eligible. In the case of SSI, as of its last adjustment in 1989, enrollees cannot have savings of more than $2,000 as an individual or $3,000 as a family. Furthermore, SSI beneficiaries are prohibited from having retirement accounts, life insurance policies, certain types of personal property, funeral/burial policies, and access to other types of income.
[emphasis mine]
OMG I'm gonna test some means! 
It's me!
I actually got approved on my first try, without a lawyer or anything. I had many years of medical records, and also had to go to one of their affiliated doctors for a final check-up.
There is one way you can have savings, but you have to have had your disability before the age of 26, and you're only allowed to spend it on certain things. Also your Social Security caseworker will not have heard of this program and you'll just have to hope they don't make it count against you somehow. https://en.wikipedia.org/wiki/ABLE_account
Another fun fact - if someone buys you dinner, you have to report it, so they can reduce your benefits accordingly.
When benefits are raised or reduced, that happens two months later, so if you managed a desperate attempt to make some money or received assistance, it's dangerous to spend it on what you needed it for, because at ~$8 a day after rent you can't really afford to let that go down two months from now when prices might be even higher than they are now.
Also, you can't pay less than an even share of rent, even if you send them a blueprint of your apartment showing that you have the smallest bedroom, because anything less than an equal split counts as "housing assistance." Your benefits will be slashed accordingly.
There's probably a dozen other restrictions you can add onto this because the list never ends, but one that got me was that you can't get paid while traveling outside the country. Every day you're gone, that's a day they slash off your month's pay. People will ofc say if you're so desperate to be on SSI you shouldn't be taking leisure trips, but this applies to people literally just crossing the border to visit family in Canada or Mexico. And they definitely keep track of when and where you're going and how long you were there. So there's an extra element of social isolation if you happen to be from an immigrant family.
I didn't know that. I thought it only mattered if you were out of the country for 30 days or more. Here's what the FAQ says:
Excerpt
If you leave the United StatesLeaving the United States means leaving the 50 states, the District of Columbia, or the Northern Mariana Islands. Usually, if you leave the United States for 30 days or more, you can no longer get SSI.
If you move to Puerto Rico, you’re considered to be outside the United States for SSI purposes only. People who live in American Samoa, Guam, Puerto Rico, and the U.S. Virgin Islands cannot receive SSI.
If you plan to leave the United States, tell us before you leave. We need to know the date you plan to leave and the date you plan to come back. Then, we can tell you if your SSI will be affected.
After you have been outside the United States for 30 or more days in a row, your SSI can’t start again until you have been back in the country for at least 30 straight days. There are special rules for dependent children of military personnel who leave the United States. They may be able to get or apply for SSI while overseas. There are also exceptions for students studying abroad.
Source (warning: PDF). After what you've said, that "Then, we can tell you if your SSI will be affected" sounds especially ominous, but this page seems to indicate that benefits aren't affected if the time out of the country is less than thirty days.
Empire moment.
"Now sign up for our armed forces."
It honestly sounds like something they'd do, so I'll get a firm answer from a case worker beforehand if I ever have the chance to leave the country. At least I know for sure that although I can't leave my state for more than thirty days (given the state's portion of the benefits), they don't dock me when I'm elsewhere.