Is there a way to estimate beta from regression analysis other than the mle estimation? Is there a way to estimate beta from regression analysis other than the mele estimation?
Or is there any other distribution regression where the closed form solution exists other than normal regression?

Only under an assumption of a normal error term does ordinary least squares coincide with maximum likelihood estimation, so if you assume a different distribution of the errors (or make no assumption at all) yet run through the usual $\hat\beta_{ols}=(X^TX)^{-1}X^Ty$ calculation, you’re not doing maximum likelihood estimation.
Even if you do assume a Gaussian error term, there are alternatives to the ordinary least squares (equivalent to maximum likelihood) estimator. One such option is ridge regression, which does have a closed-form solution. Let $I$ be the identity matrix with the same dimensions as $X^TX$, and let $k>0$ be a number of your choosing (perhaps by some kind of cross validation to find the value that gives the best out-of-sample cross validation performance). Then the ridge regression estimator is:
$$
\hat\beta_{ridge,k}=
(X^TX+kI)^{-1}X^Ty
$$
Note that the ridge estimator is a biased estimator. While “bias” as a technical term in statistics does not have the negative connotations that the word often has in colloquial English, it is worth being aware of this fact.
