Someone asked on MSE about an upper bound for $x$ in $x^2 - p y^2 = 1,$ when $p$ is prime. I checked, it appears reasonable to guess that $$ x < p^{\sqrt p} $$ when $p > 2.$ I had the computer solve by Lagrange's method, no continued fractions, no decimal accuracy required, no memory required, but the method is still elementary. I had the machine print out whenever $\log_p(\log_p(x))$ increased. It was necessary to take $p > 2$ because $x=3$ gives an overly large logarithm. Meanwhile, if all we do is print whenever $x$ itself increases, there are several composite numbers below $100$ that get included, after that they give way to primes $p \equiv 1 \pmod 4.$

So, the questions would be, (I) what is unconditionally proved about the size of $x,$ (II) what is proved under conjectures that people mostly believe true, (III) what are the most optimistic things conjectured?

p                                       
5        log_p(x)     1.365212388971971   log_p(log_p(x)) 0.1934277864616169   X 9  
13       log_p(x)     2.524585016802303   log_p(log_p(x)) 0.3610506760085375   X 649  
61        log_p(x)    5.17947382679923   log_p(log_p(x))  0.4000860954668999   X 1766319049  
109      log_p(x)     6.969012778576543   log_p(log_p(x)) 0.4138413148682316   X 158070671986249  
421      log_p(x)    12.79922341582056   log_p(log_p(x))  0.4218996203501611   X 3879474045914926879468217167061449  
1621     log_p(x)    23.61505725662223   log_p(log_p(x))  0.4278136548619654   X 6298101812493732343034974500091457815529942308667051412857352310169665125001  
.....................
44450701  log_p(x) 2641.408511213517     log_p(log_p(x))  0.4474228404332914   X  is rather large...