A user on MSE, @martin , asked https://math.stackexchange.com/questions/1611411/pell-equations-upper-bound 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 $p \equiv 1 \pmod 4.$ I put in a fair amount of effort but was unable to draw any firm conclusions.

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...

Why not, here is how it begins if we print every time $x$ increases and make no requirement about loglog, allowing $n$ composite in $x^2 - n y^2$ with $2 \leq n \leq 500$

2  log_p(x) 1.584962500721156   log_p(log_p(x))   0.6644487074538893   X 3
5  log_p(x) 1.365212388971971   log_p(log_p(x))   0.1934277864616169   X 9
10  log_p(x) 1.278753600952829   log_p(log_p(x))  0.1067868696893203   X 19
13  log_p(x) 2.524585016802303   log_p(log_p(x))  0.3610506760085375   X 649
29  log_p(x) 2.729264122987999   log_p(log_p(x))  0.298171610554983   X 9801
46  log_p(x) 2.637925539730376   log_p(log_p(x))  0.2533517055829028   X 24335
53  log_p(x) 2.79606031271967   log_p(log_p(x))   0.258976271165875   X 66249
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
181  log_p(x) 8.146702019142648   log_p(log_p(x)) 0.4035037766708247   X 2469645423824185801
277  log_p(x) 8.271023203635528   log_p(log_p(x)) 0.3756670785256742   X 159150073798980475849
397  log_p(x) 8.05129073299257   log_p(log_p(x))  0.3485719633766078   X 838721786045180184649
409  log_p(x) 8.576275777667302   log_p(log_p(x)) 0.3573497754649824   X 25052977273092427986049
421  log_p(x) 12.79922341582056   log_p(log_p(x)) 0.4218996203501611   X 3879474045914926879468217167061449