I have worked in higher education in various U.S. states and several other countries as both a professor and an academic adviser. In the latter case, my job was to help students "navigate the system." Getting through college and paying for it can be complicated if you are trying to do it alone. It's worth learning how to pay the game and how to find help getting through.
A few thoughts:
Finding merit-based scholarships as a non-citizen is not easy. The key is to go local. Funding from your department, school, or local community groups and businesses is much less likely to require any proof of legal status. Most scholarship committees frankly don't care. I have worked as a scholarship judge. The last thing you want is extra info from a thousand applicants that won't help you choose the best people. However, such scholarships are usually small, in the $500 - $2,000 range, and are usually awarded after you have already been enrolled in a certain school. The ones that pay out more have more applicants and are a long shot. These will be a bonus but not a way to pay most of your tuition. Sometimes it's easier just to work than take the time to apply to the smallest awards.
Besides scholarships, another thing to think about is the price tag of the university. Costs vary widely among states. Community colleges are much cheaper than four-year schools and I would strongly recommend starting out at one. It's easy to find a good community college. Pick an upscale suburban neighborhood, a top-tier university area, or liberal arts college town and find the nearest community college. Ivy-type schools usually overshadow a really good community college. The bonus is that you can hang out with the top students from the four-year school while paying cheap tuition. You might even be able to build a support network that helps you transfer into the four-year school. Since you're undocumented, lean toward areas with large numbers of undocumented students, or at least places that are not hostile to immigrants. To state the obvious, Arizona and Texas would not be first on my list. Places that tend to spend more on public infrastructure and have rich people hanging around that give money away are probably a good bet, too. I'm guessing you won't find as much financial largess in the South.
Among U.S. states, California seems to have the most tolerant university system for undocumented students I've seen, probably partially because there are so many in the system and because everyone is related to someone, or at least has a friend, who is undocumented. They have their own category in the system: AB 540.* Even though you would not be legally considered AB 540, I think you could just tell people you are when interacting with college staff and faculty. If "undocumented" is a nice way to say "illegal," "AB 540" is a nice way to say "undocumented." Plus, it protects you because the sort of person who would turn you in doesn't know what it means. Unlike other states I've been in, students openly discuss their status with college staff, professors, and classmates in CA. I have talked with admission and registration staff at several urban colleges about this issue. They are adamant that they do not and will not share information on legal status with any branch of the U.S. government. Always be careful, but you're better off in this kind of environment. For example, I overheard a student approach an engineering department chair about the fact that he couldn't apply for most big corporate internships because of his status. The professor advised the student to continue studying as long as he could (i.e. to the PhD level) and get top grades. He said that becoming indispensable in some technical field will earn you a visa in the U.S. or a good job in some other country. Not sure if that's accurate or good advice, but it was interesting that the student felt comfortable approaching a professor he didn't know well about this problem, and the professor addressed it completely matter-of-fact-ly.
Undocumented students, even those from middle class families, tend to be impoverished because they have to pay more tuition and work very low paying jobs. In general, I have noticed that California college administrative staff tend to be more understanding of very low income students than staff in other states. For example, they will make arrangements for installment payments of smaller sums of money. The community college system allows for paying bills later in the term as well. In some states, the computers automatically drop your classes if you haven't paid after the first two weeks, but I believe this is illegal in CA. In the CA State University (CSU) system, annual in-state tuition is now around $6,000 a year, a bargain compared to many other states, but considered expensive by locals. Out-of-state tuition is considered extremely expensive, which means you will get more help and sympathy trying to pay your tuition. Increases in tuition are happening regularly due to the budget fiasco, but having sympathetic people to help you through is worth a lot. Often, staff control scholarship dollars locally and have extra money to allocate. Most students don't know this, but when not enough people apply for money dedicated to African-American and Latino students, it can often go to you if you ask for it. Getting to know these staff people is worth a lot. They often have titles such as academic advisor, MESA coordinator, TRIO--there are lots of acronyms that just refer to different pots of federal money that pay their salaries, but they are usually housed in "Student Affairs" or "Student Services." The key web search terms related to these people are "students at risk," "disadvantaged" or "first-generation" students. "Leaving college" is the PC term they use for dropping out. When I did this job, my university had a row of five or six offices that each did the same thing, but our salaries all came from different funds. Any of us would have helped you if you had asked. Many officially support a certain ethnicity--according to their job description--but if you ask for their help, they will help you regardless. If you are familiar with the concept of a "case manager," they do something similar. Their job is to know about where money and resources are, and which people in the rest of the university (financial aid/registration/etc.) are helpful and which should be avoided. Pick a region where you want to study and start visiting these people. Just cold call them and walk into their offices. You will eventually find someone who will help you out, and that's a good place to go to school.
If this doesn't work out and you do end up deciding to leave the country, you have a few things going for you.
To increase their international rankings and sometimes to meet local political initiatives, universities in many countries are trying to ratchet up the number of foreign students they accept. Most of them can't find enough people from "rich" countries. Some of them will give you a free ride so that their "international program" is slightly more international and not all Chinese and Pakistani students. Search online for "international studies" or "international business" programs.
Top universities in some countries would love to have you just because you are "American" to them (the irony) and speak English. You are guaranteed to get into most of the top schools in South Korea, for example, and then be guaranteed lab research opportunities with whomever you ask, if you are interested in science or engineering. Your degree will have much more value inside the country than outside it, but if you are actually interested in the place where you study, it might be worth it.
Since you're from Saudi, you might consider the Middle East. There's loads of money in new universities in places like the UAE and they want international students. The money still exceeds the quality, but they are intent on spending to get the quality up, and money is what you're looking for. Go there for undergrad, then get a free graduate (or "post-graduate" as they call it) degree in Europe.
Some countries still have cheap or free tuition. Look at Scandinavian schools. Look at the top schools in Latin America, such as UNAM in Mexico City. Many of these schools will have extensive course offerings in English, although I recommend learning the local language on your way through school. Heck, maybe you could even stay in the U.S. and figure out how to study online at a foreign university, although you'd probably have to pay cash. Check out Australia for that. They have been rapidly expanding their higher education system internationally.
I guess I could go on, but this should get you started thinking about these things. A couple of websites that will give you a few more search terms:
* http://ab540.com/WhatIsAb540.html
http://www.californiacolleges.edu/finance/financial-aid-faqs...