First, I'm not a mathematician; my hobby is mainly memorization. I want to practice math formulas and/or equations memorization. In that way, I'm looking for large and complex formulas or equations for memorization practice, nothing for academically goals.
So, what are your suggestions? What formula or equation do you believe is large and complex and represents difficult memorization?
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"What formula or equation do you believe is large and complex and represents difficult memorization?"
I am describing below approaches of generating expressions that look complex but you could have some control over the level of complexity.
1-You could generate complex expressions using the expansion of $(f(x)+g(x)^n$ by applying the Binomial Theorem. the result gets long and complicated for large values of n and non-trivial g(x) and f(x).
For example, you could try $(cos(x)+(x^3+x+1)sin(x))^{10}$.
2-You can try expanding a function of any complexity using Taylor series. You will end up with really long and fancy looking equations.
3-Multiply functions by one another, for example $(cos(x)+sin(x)+tan(x))(1+x+x^2+...+x^5)$. Again, you can choose the functions and produce very complex looking equations.
There are calculators that show the value for the above, you don't have to manually work it out, it is not a lot of fun!
Edit
I have just came across this, in case you are interested. Source is in
Wiki-Si(x)-Ci(x)
References:
Here is a very personal and partial answer.
["Disclaimer"] Among students in mathematics, or persons who do, more or less professionaly, mathematics, you will scarcely find people who rely heavily on memory, compared to other domains like chemistry, biology, not to speak about medicine studies... The training in mathematics encourages to remember methods more than raw results under the form of formulas.
Besides, one could built arbitrarily untractable and useless mathematical formulas as evidenced in the answer by @NoChance.
This said, it is helpful to rely on memory for useful results like the ones (chosen among others) that I am going to enumerate (I am confident that any mathematician can consider all of them as important) :
$$\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b) \tag{1}$$
$$a^2=b^2+c^2-2bc \cos(A)\tag{2}$$
(generalized Pythagorean relationship)
$$B(a,b)=\int_0^1 t^{a-1}(1-t)^{b-1}dt \tag{3}=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)}$$
(Beta integral : https://en.wikipedia.org/wiki/Beta_function)
$$\Gamma(x)\Gamma(1-x)=\frac{\pi}{\sin(\pi x)}\tag{4}$$
(complements' formula)
$$\frac{\partial f}{\partial t}=k \frac{\partial^2 f}{\partial x^2} \tag{5}$$
(Diffusion equation : https://www.uni-muenster.de/imperia/md/content/physik_tp/lectures/ws2016-2017/num_methods_i/heat.pdf)
These formulas (1) to (5) will be, according to two different persons, easy or not to memorize, as a whole. A certain rythm, I would say, a certain music, or a certain balance, will help to memorize formula (3) or formulas (4). Formula (5) can be memorized as presenting a symmetric-assymetric aspect : partial derivatives on both sides, but first order on one side and second order on the other ; but which one is with respect to $t$, which one with respect to $x$ ?
A (rather classical) division exist between mathematically inclined : you will find among them
visual people (good in particular at geometry),
auditory people (who prefer algebraic methods, and in particular formulas, here we are, even if is an oversimplification) and
kinesthetic people (to a certain extend, it corresponds to people who will be impatient to program a mathematical concept in order to have a full grasp on it).
But this distinction can be futile for some formulas that have a geometrical aspect like
$$\Phi=1+\frac{1}{1+\frac{1}{1+\frac{1}{1+...}}}$$
(continuous fraction decomposition of the golden ratio).
Now, which formulas would be especialy difficult to remember ? The length criteria (taking length = number of symbols) shouldn't be retained as such. The main criteria of difficulty is the lack of immediate connection between LHS and RHS of a formula.
At the foremost place for the difficulty of memorization are Ramanujan formulas (as @Felix Marin has pointed to) ; they are stunning by the unexpected connection they make with very different parts of mathematics (https://faculty.math.illinois.edu/~berndt/articles/aachen.pdf).
Personaly, I find difficult to remember :
$$|x|=\frac{\pi}{2}-\frac{4}{\pi}\sum_{n=1}^{\infty}\frac{1}{(2n-1)^2} \cos((2n-1)x)$$
https://proofwiki.org/wiki/Fourier_Series_of_Absolute_Value_of_x
(take $x=0$ to obtain a classical formula for $\pi^2$).
$$\theta_1(x+y|\tau)\theta_1(x-y|\tau)\theta_2(u+v|\tau)\theta_2(u-v|\tau)=\theta_3(y+u|\tau)\theta_3(y-u|\tau)\theta_4(x+v|\tau)\theta_4(x-v|\tau)-\theta_3(x+u|\tau)\theta_3(x-u|\tau)\theta_4(y+v|\tau)\theta_4(y-v|\tau)$$
(https://msp.org/pjm/2009/240-1/pjm-v240-n1-p05-p.pdf), Bessel functions like this one I have been working on recently : https://math.stackexchange.com/q/3063945 or definitions of orthogonal polynomials like associated Legendre polynomials,
$$(1+UV^T)^{-1}=A^{-1}-A^{-1}U(I+V^TA^{-1}U)V^TA^{-1}$$ (https://en.wikipedia.org/wiki/Sherman%E2%80%93Morrison_formula) and the Cayley-Menger determinant https://en.wikipedia.org/wiki/Cayley%E2%80%93Menger_determinant,
$$\displaystyle \prod _{n=1}^{\infty }\left(1-x^{n}\right)=\sum _{k=-\infty }^{\infty }\left(-1\right)^{k}x^{k\left(3k-1\right)/2}=1+\sum _{k=1}^{\infty }(-1)^{k}\left(x^{k(3k+1)/2}+x^{k(3k-1)/2}\right).$$ (https://en.wikipedia.org/wiki/Pentagonal_number_theorem) and still others like Euler-MacLaurin formula :
$$\displaystyle \sum _{i=m}^{n}f(i)=\int _{m}^{n}f(x)\,dx+{\frac {f(n)+f(m)}{2}}+\sum _{k=1}^{\lfloor p/2\rfloor }{\frac {B_{2k}}{(2k)!}}(f^{(2k-1)}(n)-f^{(2k-1)}(m))+R_{p}.$$
Remark : @user295959 mentions the memorization of the first dozen of decimals of special numbers like $\pi$. This memorization can be done by remembering some associated sentences with mnemonics (words' lengths = decimals) as one can find in https://www.ict4us.com/r.kuijt/en_pi_onthouden.htm, but the OP considers this out of scope.