How do I generate a list of non-repeating random integers…

So you have to track your selections, then select another non-repeat. In any programming language you can use an array, but you would always use a loop, for or while whichever. In Excel cells can act as loops, and they are your variable storage, however the programatic functionality does not exist like C/C++, Java, R, Matlab/Octave, Perl, Python, Ruby. It is much more similar to Scheme (although Scheme uses no variables at all, but it's the concept of return value being the only storage available). So when you remove the ability to use arrays, VBA code (so no loops programatically), you are left with a circular reference option of a simulated loop, however it has a limited number of iterations to prevent infinite loops, which means a solution is not guaranteed in a algorithmic mean, so it will not work in all cases. To do what you are asking in Excel, the only remaining option is a VBA function saved in an xlsb (excel binary file) that excel can use, but only on machines with that file. To solve the problem you need one of the tools you have eliminated.

I'm not sure how many randoms you are drawing without repeats between 1 and 10, but I am attaching the iterative (circular) method here. I have the probabilistic numbers shown for the number of iterations (which can be controlled in Excel options), the max is 32767, which is what it is set to (so it will circularly calculate up to 32767 times for one cell). I have manual calculations turned on with recalculate on save turned off. So you can press F9, which randomizes the seed values in column A, which drives column B to check for repeats, if it finds a repeat it draws another random, then checks itself verse all randoms above it (so the second checks the first, third value checks the first and second). It stops when unique, or when it hits the max number of iterations (that's the part that eliminates the globally optimal solution, but if you in all case is statistically less than 1%, although with 8 randoms I got 12% in practice, and about 2% had a repeat at the end of run with 6 randoms). This is because statistical probability assumes a perfect uniform random distribution to draw from, which no computer has, so the statistical probability is the best case. That's why I added the checks in column C, it will say repeat next to the repeating values. So you would just press F9 again. I'm not sure if this is a toy example for you and you are going to expand it or what you are finally doing with the product but if you can elaborate I might be able to make it work a little better. I don't want to bother you, but I had time on the plan and in the Sky Club, so I looked into it a little more (max iterations were 1000 last time I had looked, so the 32767 makes a difference in practice).