I've recently been studying about math, and when learning about decimals. I found myself with this question.
An explanation I got from Quora was: Arithmetic operations such as addition, subtraction, multiplication, and division would have to be adapted to the new system. For example, the sum of two numbers that would currently result in 20, in the new system would result in 10-10.
But, I want to know if there are more examples where 10 as a unique digit would make harder the operations. Thanks.
On
One nice thing about a base system as it is usually configured is that representations of integers are unique. In your nonstandard base ten system, where you have a digit representing ten, let's call it $T$, you no longer have unique representation of integers. For instance, ten could be represented as $T$ or as $10$ ($1$ ten and $0$ ones). Or one hundred could be represented as $T0$ or $9T$ or $100$.
It is perfectly possible to consider base $N$ number representations which don't restrict digits to the range $0 \ldots N-1$. Allowing the digits to range from $0$ to $10$ in a decimal system makes the system ambiguous: writing $A$ for the digit $10$, you can write twenty as either $20$ or $1A$ (not $AA$ as you suggest). It is a moot point whether this makes the arithmetic operations harder: on the one hand, there are more digit-wise operations to think about, but on the other hand you can avoid some carries and borrows: $9 + 1 = A$ with no carry.
If you are interested to know more about this, then Volume 2 of Knuth's Art of Computer Programming is a good place to start. One of the fascinating systems that Knuth discusses is balanced ternary: base $3$ arithmetic using the digits $-1$, $0$ and $1$.