How to evaluate powers of powers (i.e. $2^3^4$) in absence of parentheses?

Barring parentheses, $2^{3^4}$ should definitely be read as, and is equivalent to $\;2^{\left(3^4\right)}$:$$2^{3^4} = 2^{(3^4)} = 2^{81}$$ whereas $${(2^3)}^4 = 2^{3\cdot 4} = 2^{12}$$

Added:

As pointed out in the comments, it is fairly standard practice that exponents are "right associative" - which is somewhat of a misnomer which should only be taken to mean, evaluate rightmost first: read a^b^c as a^(b^c) and read a^b^c^d as a^[b^(c^d)], and so on. As the example above shows,the exponential operator is not associative. So as you have indicated you typically do, use parentheses, when possible, in your own usage, to avoid any possible confusion.