Given an iid sample $X_{1}, \ldots, X_{n} \sim Bin(n, p)$ I'm trying to find
$$E(X_{(n)})$$
that is the expectation of the sample maximum. Unfortunately I don't know where to start. It seems that the maximum does not follow a Binomial distribution, since its CDF is given by the product of the CDFs of the observations which is not equal to the CDF of a Binomial distribution.
Is there a "trick" one can use to find the expectation (relatively easily)?
For a random variable whose values are nonnegative integers, $E[X] = \sum_{x\ge 0} (1 - F_X(x))$. So in your case $$ E[X_{(n)}] = \sum_{x=0}^n\left(1 - F(x)^n\right) = n+1 - \sum_{x=0}^n F(x)^n $$ where $F$ is the CDF of the binomial distribution. But I don't think there's a nice closed form. The first few cases are $$ \matrix{n=1 &p \cr n=2 &-2\,{p}^{4}+4\,{p}^{3}-4\,{p}^{2}+4\,p \cr n=3 &-6\,{p}^{9}+27\,{p}^{8}-18\,{p}^{7}-72\,{p}^{6}+162\,{p}^{5}-153\,{p}^ {4}+81\,{p}^{3}-27\,{p}^{2}+9\,p \cr n=4 &-164\,{p}^{16}+1312\,{p}^{15}-5088\,{p}^{14}+12656\,{p}^{13}-21936\,{p }^{12}+26592\,{p}^{11}-22048\,{p}^{10}+13104\,{p}^{9}-8376\,{p}^{8}+ 8416\,{p}^{7}-7840\,{p}^{6}+4944\,{p}^{5}-2032\,{p}^{4}+544\,{p}^{3}- 96\,{p}^{2}+16\,p \cr n=5 &5730\,{p}^{25}-71625\,{p}^{24}+365500\,{p}^{23}-908500\,{p}^{22}+ 544400\,{p}^{21}+3886950\,{p}^{20}-15577750\,{p}^{19}+32983750\,{p}^{ 18}-47818625\,{p}^{17}+50498650\,{p}^{16}-38593350\,{p}^{15}+19088000 \,{p}^{14}-2268000\,{p}^{13}-6031250\,{p}^{12}+6803900\,{p}^{11}- 4418950\,{p}^{10}+2184375\,{p}^{9}-974125\,{p}^{8}+430500\,{p}^{7}- 175100\,{p}^{6}+57125\,{p}^{5}-13625\,{p}^{4}+2250\,{p}^{3}-250\,{p}^{ 2}+25\,p \cr }$$
