Given a polynomial $P$ with integer coefficients in finitely many variables, we denote by $v(P)$ the product of the absolute values of the non-zero coefficients and the non-zero total degrees of the monomials of $P$.
Since there is no general algorithm to decide whether an equation $P(x_1, \dots, x_k) = 0$ has a solution in integers $x_1, \dots, x_k$ or not, for every computable function $f$ there is a polynomial $P$ such that the equation $P(x_1, \dots, x_k) = 0$ has such solution, but none for which all $x_i$ have absolute value less than $f(v(P))$. In particular, this holds for the function $f: n \mapsto 2^n$.
For the purposes of this question, we say that the diophantine equation $P(x_1, \dots, x_k) = 0$ has large smallest solution if it has a solution, but none such that all $x_i$ have absolute value less than $2^{v(P)}$.
Question: Which are ''nice'' examples of diophantine equations with large smallest solution?
Here are some non-examples:
Put $P := x-1$. Then, $v(P) = 1$, but the equation $P(x) = 0$ has the solution $x = 1$ whose absolute value is less than $2^{v(P)} = 2^1 = 2$.
Put $P := a^2 + b^2 - c^2$. Then, $v(P) = 2^3 = 8$, hence our bound is $2^8 = 256$. Even not counting the solution $a = b = c = 0$, there is e.g. the solution $a = 3, b = 4, c = 5$ where all values are less than $256$.
Put $P := x^2 - 61y^2 - 1$. Then, $v(P) = 2^2 \cdot 61 = 244$, and the solution $(x,y) = (1766319049,226153980)$ is clearly less than the bound of $2^{244} > 10^{73}$.
Put $P := a^4 + b^4 + c^4 - d^4$. Then, $v(P) = 4^4 = 256$, hence our bound is $2^{256}$. Even not counting the solution $a = b = c = d = 0$, there is e.g. the solution $a = 95800$, $b = 217519$, $c = 414560$, $d = 422481$, where all values are less than $2^{256} > 10^{77}$.
Put $P := x^3 + y^3 + z^3 - 33$. Then, $v(P) = 3^3 \cdot 33 = 891$, and the absolute values of all numbers in the solution $$ (x,y,z) = (−2736111468807040,−8778405442862239,8866128975287528) $$ are clearly less than the bound of $2^{891} > 10^{268}$.
Now, let's have a look at the equation $$ \frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} \ = \ 4. $$ This truely has a large smallest solution, at least if one insists on $a$, $b$ and $c$ being positive, hasn't it? —
No, not really: subtracting 4 from both sides of the equation, and multiplying the equation by the common denominator of the terms, we arrive at the polynomial $$ P = a^3-3a^2b-3a^2c-3ab^2-5abc-3ac^2+b^3-3b^2c-3bc^2+c^3, $$ which has $$ v(P) = 3 \cdot 3^2 \cdot 3^2 \cdot 3^2 \cdot 3 \cdot 5 \cdot 3^2 \cdot 3 \cdot 3^2 \cdot 3^2 \cdot 3 = 215233605. $$ We observe that the smallest solution even doesn't exceed $2^{267}$, let alone $2^{215233605}$.
Addendum: In order to further explain what I mean by ''nice'' in this context: while it is straightforward to build equations which in some obvious ways encode computations which yield large numbers, the art is rather to find equations which do not do so in any obvious way (like the non-examples above), but which still satisfy the condition on the size of the smallest solution. — And these latter equations are rather the ones which are ''nice'' in the sense of this question. Nevertheless, in case no good such examples are found, also the equations of the former type are valid answers to the question (just not such good ones).