If I start with no current license, can I take all the exams in one sitting to get to Amateur Extra?
Yes, you can take all three exams in sequence at the same session and obtain an Extra class license. (I did and had no trouble.)
Doing this, rather than getting a lower license class and upgrading later, has the small advantage that you will be assigned a shorter sequential call sign (without having to request one when you upgrade), as well as not paying additional exam session fees.
However, I would recommend that you arrive early, and inform the VEs that you are planning to do this. I've heard that in some exam sessions, the examiners qualified to administer the General and Extra exams (that is, ones who themselves hold Extra class licenses) may leave after everyone who came to upgrade their licenses has taken their single exam, whereas you will be starting only after you finish your Technician exam.
After you pass an amateur exam, you will receive an CSCE (Certificate of Successful Completion of Examination). This proves that you have passed the exam. This ticket also allows you to take the next exam. In recursion, a new ham can go from having no license to obtaining an Extra in one sitting... for the same price as your original technician exam.
In 1995, I was living in the US for three months. I bought some study books, read all the question pools and booked myself in to the local test centre. I had planned to take the 20wpm Morse test, and if I failed that then the 13wpm test, and if for some reason I failed that then the 5wpm test. Then I had planned to sit the five exams for Novice, Technician, General, Advanced and Extra classes of licence. As it happened, I scraped a pass in the 20wpm Morse test and passed all of the other exams (the Advanced test was the hardest! The pool was enormous), and came out with pass slips to allow me to apply for an Amateur Extra licence, the highest there is.
All in all, it was a good morning's work! The guys at the test centre said I was the first person to manage that at their centre, although they had heard of people doing it before.
I did also at that time hold a UK class A licence, otherwise I would have been extremely nervous at the prospect of taking between 6 and 9 tests for a radio licence!
You'd have to take all three "written elements." From the FCC Licensing Examinations page:
In the amateur radio license examination system, there are three written examination elements [Elements 2, 3, and 4]. For each operator class, you must pass the elements indicated:Operator Class Required Elements Amateur Extra Written Elements 2, 3, 4 General Written Elements 2, 3 Technician Written Element 2
The written elements are:
Element Operation Privileges Questions Minimum Passing Score 2 Technician Class 35 26 3 General Class 35 26 4 Amateur Extra Class 50 37