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: 
That's a good question. In addition to PEMDAS there's grouped symbols; they occur under a radical ($\sqrt{~}$, $\sqrt[n]{~}$), in the numerator and denominator of a fraction and in exponents. These grouped symbols are treated as if they are between parenthesis. That means that the expressions $a^{\text{expression}}$, $\sqrt[n]{\text{expression}}$ and $\dfrac{\text{expression}\_1}{\text{expression}\_2}$ mean $a^{(\text{expression})}$, $\sqrt[n]{(\text{expression})}$ and $\dfrac{(\text{expression}\_1)}{(\text{expression}\_2)}$.

The horizontal fractional line and the [vinculum](https://en.wikipedia.org/wiki/Vinculum_(symbol)) of the radical act as symbols of grouping. That's why we write $\sqrt{\text{expression}}$ instead of $\sqrt{~}(\text{expression})$ and $\dfrac{\text{expression}\_1}{\text{expression}\_2}$ instead of $(\text{expression}\_1)/(\text{expression}\_2)$. The exponent, written in superscript, is also a grouped symbol, so we write $a^{\text{expression}}$ instead of $a\wedge(\text{expression})$.

Other functions use parentheses around the input to avoid ambiguity. The parentheses are sometimes omitted if the input is a monomial. That doesn't have anything to do with units. Thus $\sin 2x$ means $\sin (2x)$ but $\sin a+b$ means $\sin (a)+b$. Personally I prefer to write the parenthesis even around a monomial like $\ln (5x)$. 

What does $a/bc$ means? Is it $\dfrac{a}{bc}$ or $\dfrac{a}{b}c$ ? We could argue which convention we like but the matter of fact is that different people have different convention and different softwares have different conventions. For example $18/2\times 3$ gives [$27$](https://www.wolframalpha.com/input/?i=18%2F2*3) on wolframalpha but $a/bc$ gives [$3$](https://www.wolframalpha.com/input/?i=a%2Fbc+where+a%3D18+and+b%3D2+and+c%3D3) when $a=18$, $b=2$ and $c=3$. The same software has two conventions for the same expression! That's why I always prefer to write $\dfrac{a}{b}$, $\sqrt{a}$ and $a^b$ instead of $a/b$, $\sqrt{}a$ and $a\wedge b$. They don't cause any ambiguity. The same issue is present with $\sin ab$ which WA interprets as [$b\sin a$](https://www.wolframalpha.com/input/?i=sin2*3) or [$\sin (ab)$](https://www.wolframalpha.com/input/?i=sin+ab+where+a%3D2+and+b%3D3).
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 $x$^$y$^$z = ((x$^$y)$^$z)$