How to write Superexponents

If you nest several super/subscripts, you should disambiguate the writing since it is not clear (for LaTeX) if you want to write (2^2)^65533 or 2^(2^65533).
You should therefor use curly brackets around the argument of the power function for having an unequivocal notation.

In your case, you should write $2^{2^{65533}}$

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_{(note that $2^{2^65533}$ is also ok, since it is already unequivocal)}

The right use of the exponent function is ^{<argument>} (e.g. $e^{i\pi}$).
However, if your argument is one character-long only, you can use the shorthand ^<argument> (i.e. without curly brackets, e.g. $2^4$). Thus, you cannot write $2^{2^65533}$, even if it's mathematically unequivocal.

The LaTeX manual always shows the syntax

<base>^{<exponent>}

You should type

10^{2}  e^{x}  e^{-x^2}  \sum_{k=0}^{\infty}\frac{x^{n}}{n!}

If you always do like this, your input would be

2^{2^{65533}}

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If you type 2^3^4, you get the double exponent error message, because the input is ambiguous. Other software might accept it, usually interpreting it as if it were 2^81, but TeX is a typesetting system, so it wants you to be clear about your intentions.

Most importantly, 2^81 would be ambiguous as well, again because TeX is a typesetting system and cannot distinguish 2^8x and 2^8 x.

CAS software might be able to interpret 2^8x as “x multiplied by 256”, TeX does not interpret anything because it doesn't do math.

If you want “2 to the exponent 8x”, type 2^{8x}; if you want “256 times x”, type 2^{8}x.