How much do top tier Universities expect their students to know beyond the syllabus?

Top Tier as in: Cambridge, Oxford, Imperial College, Warwick, Durham, Bath, UCL sort of level

How much am I expected to know beyond the A level maths syllabus? (I'm doing modules C1-C4, M1, M2, S1 this year for AS, and M2-M4, S2-S4, FP1-FP3, Differential Equations, Decision Maths for A2).

Anyway, I can't be bothered to focus on my incoming exams in June at the moment so I'm deciding on what else Mathsy to do that is productive.

To get into these top tier universities should I start
Start studying FP1-FP3 now
Learn the lemmas and techniques that are useful for Olympiad type problems
Learn more about my favorite area of maths (Number Theory), and as many proofs and related topics as I can? (e.g. Wilson's Theorem, The common Prime counting estimate functions such as Li(x), Gamma and Zeta functions, Reciprocals of primes, Prime/Integer facts (a lot from Fermat such as any prime of the form 4k+1 is expressable as the sum of 2 squares, Lagrange's theorem, Eulers Totient theorem) (I haven't learnt any of the proofs for the above listed yet)

Will they expect me to know at least some of the above?

They won't "expect" you to know any of those things; however, people who are aiming for those sorts of university have typically read widely outside the A level syllabus and can demonstrate that they're not put off by interesting and challenging problems that require concentration and a lot of hard work!

But your top priority is to get some demonstrable results on paper - make sure you definitely aren't going to screw up in your AS exams first, then you have plenty of time to explore the more interesting aspects of mathematics. If you're happy that you've got your 1st year modules nailed, then move on to the FP1-FP3 modules early because these are where things get more interesting. Once the A level concepts have matured in your brain a bit more, you'll be in a position to tackle some more challenging problems like AEA and STEP exams - have you come across these yet?

I honestly cant be assed to study more for my AS levels at the moment. The material for the exams is really quite basic, and what the exams are really testing are how careful you can be, and although I'm only getting about 90-95% in terms of marks for the mocks I do in class, I do tend to finish some 30 minutes early for most papers and don't bother to check and just leave because it's a mock and lose marks due to careless errors.
I had a look at STEP exams. They appear to be somewhere between BMO1 and BMO2 in terms of difficulty for each question and ingenuity required, but the syllabus that you are expected to know is signficantly wider.
Is training for these STEP exams early more productive than reading up all the facts and proofs I can on number theory? because that's what I'm doing at the moment.

In terms of doing things that are "interesting" there's no right answer to your question - I wouldn't want to put anyone off focusing on the topics that interest them because I did an awful lot of it myself when I was at school! I mention the AS exams purely out of 'courtesy' because it's easy to tell people to read what they like, but if they spend all their time being distracted and neglect their formal exams then it's easy to drop marks (or even a grade) which puts them at a disadvantage later on!

I mention STEP specifically because it forms part of the offer for Cambridge and Warwick, so if you put either (or both) of those universities down as UCAS choices you are going to have to confront this exam sooner or later!

Oxford (and Imperial) use something called MAT to inform their offer process - this is taken in the first term of upper sixth (Y13) rather than at the same time as your A levels (like STEP).

STEP isn't typically required for other unis but can be used as part of an alternative offer to "trade off" against a lower A level grade e.g. you might get an offer of A*A*A without STEP or A*AA if you can get a certain grade in STEP (I think Bath operates this system).

So there are really 2 answers to your question: read whatever you want if it triggers your interest, but also be aware that there are certain exams that you have to take if you're applying to certain unis, and therefore it makes strategic sense to start investigating those options now

I've had a look at Oxford's MAT and it seems significantly easier than STEP. Am I right in saying this? I've also been told that if you get an offer from one of these MAT universities you can just "Chill" for Upper 6.

I've had a look at Oxford's MAT and it seems significantly easier than STEP. Am I right in saying this? I've also been told that if you get an offer from one of these MAT universities you can just "Chill" for Upper 6.