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
Best Answer
Let $d$ be a positive fundamental discriminant, $\epsilon_d$ denote the fundamental unit, $h(d)$ the class number, and $\chi_d$ the primitive character associated to the discriminant $d$. The class number formula gives $$ \log \epsilon_d = \sqrt{d} L(1,\chi_d)/h(d) \le \sqrt{d} L(1,\chi_d), $$ since the class number $h(d) \ge 1$. Now it is known that $L(1,\chi_d) \le C \log d$ for a constant $C$. This upper bound is completely effective. The best known constant $C$ is (for large enough $d$) $$ \frac{1}{4} \Big(2-\frac{2}{\sqrt{e}} \Big). $$ If we knew something about how small primes split in ${\Bbb Q}(\sqrt{d})$ then this could be improved by taking those Euler factors into account (for example, we can use this if $d$ is even which would happen for primes $p\equiv 3\pmod 4$ in the question). This is a result of P.J. Stephens and uses the Burgess bound for character sums (together with an argument from multiplicative number theory along the lines of Vinogradov's $1/\sqrt{e}$ argument for the least quadratic non-residue). For a discussion of Stephens's result and extensions, see Granville and Soundararajan.
This would be enough to give your conjecture of $p^{C\sqrt{p}}$ (one needs a little care to go from the fundamental unit to the solution to Pell's equation -- i.e. one might need to take a small power of the fundamental unit). Also see this paper of Hua which explicitly states a bound along the lines you want, tracing it back to Schur. Finally, Louboutin has looked at explicit upper bounds for $L(1,\chi)$ (see Theorem 5.1 there).
The above results are unconditional. On GRH one can do a bit better, since $L(1,\chi_d)$ may then be bounded by $C\log \log d$, and then one would get a better bound of $(\log p)^{C\sqrt{p}}$ in your problem (see Theorem 1.5 of this paper for an explicit GRH bound), and I think that can probably happen (although this is not clear since we don't know that the class number can get down to $1$). Jacobson, Lukes and Williams report on extensive calculations on regulators, and at the end of the paper state the belief that the fundamental unit can get as large as $\exp(c \sqrt{d}\log \log d)$; however, as also noted there, unconditionally we only know that the fundamental unit (or the solution in Pell's equation) sometimes gets as large as $\exp(c(\log d)^4)$ -- so there is a very large gap in our understanding. (See also my answer to the related MO question Upper bound for class number of a real quadratic field .)