Question: What are some real-life exceptions to the PEMDAS rule?
I am looking for examples from "real" mathematical language --- conventions that are held in modern mathematics practice, such as those appearing in papers, books, and class notes, that are not covered (or contradicted) by PEMDAS. I am not interested in contrived examples, such as those constructed [to confuse people on Facebook](https://slate.com/technology/2013/03/facebook-math-problem-why-pemdas-doesnt-always-give-a-clear-answer.html).

The PEMDAS convention says that order of operations are evaluated as:

1. Parentheses
2. Exponents
3. Multiplication and Division, from left to right
4. Addition and Subtraction, from left to right

So far I know of one exception: sometimes the placement of text indicates "implicit" parentheses: text that is a superscript, subscript, or above/below a fraction.

* $2^{3 + 5}$

Here, $3 + 5$ is evaluated before the exponentiation
* $\frac{3+5}{2}$

Here, $3 + 5$ is evaluated before the division

Are there other exceptions?

Answer: 
Briefly: The thing that you seem to be orbiting around is that some mathematical symbols, in addition to having some primary operation, also have the secondary function of serving as "grouping" symbols. These symbols serve to group certain operations together in the same way that parentheses do, without extra notation. Another way of putting it: parentheses are just one special type of grouping symbol. They do not count as exceptions to the standard precedence syntax. Examples include:

* Fraction bars
* Radical overbars
* Absolute value bars
* Superscripts and subscripts
I would add the following to Paracosmiste's reply:

Numerical negation (e.g. the '$-$' sign in in the expression $-x^2$) usually has an order of precedence that is less than that of exponentiation and greater than that of multiplication, division, addition and subtraction in order from right to left. So we have $-x^2=-(x^2)$. And $--x = -(-x))$. And $x+-y= x+(-y)$.

When exponentiation is printed using different levels of typeface (e.g. $x^{y^z}$) then they are done in order from right to left. So we have $x^{y^z}=x^{(y^z)}$ If, on the other hand, it is printed using the '^' symbol (or '$\*\*$') as in many programming languages (e.g. $x$^$y$^$z$ or x$\*\*$y$\*\*$z), they are usually done in order from left to right. So we have $x$^$y$^$z = ((x$^$y)$^$z)$