How many significant figures are appropriate when a new digit is created?
Actually, you are correct - when adding significant figures together, you keep the largest last significant figure between the two numbers. For example, $15.31+2.2=17.5$. On the other hand, when you are multiplying numbers, you keep the same number of significant figures. Using the same numbers, $15.31*2.2=34$. While this decision might appear arbitrary, there is a good reason behind it.

Since we do not know anything past the tenths place about $2.2$, lets say this number is actually $2.2x$ where $x$ is a digit that would cause this number to round down (value between 0 and 4). $15.31+2.2x = 17.5[x+1]$ (where a number in brackets is a digit that is the result of a calculation. It can carry to the next place, but for reasons that I will explain later, we assume that it does not). Since we now know nothing about the hundredths place after adding these two numbers, we cannot include it in our result.

Likewise for multiplication, $15.31 * 2.2x = 33.[6+x][8+5x][2+3x][x].$ Since we know nothing about the value of $x$, we know nothing after the decimal point. Because our next digit is guaranteed to round up, regardless of the value of $x$, we round up to get $15.31*2.2x=34$. This is not a special case. In fact - the number of digits that will not depend on the value of $x$ is always equal to the number of significant figures in that number (this is - again - because we assume that $x$ will not cause a digit to carry), thus giving us our multiplication rule for significant figures.

Now for why it is okay to assume that $x$ will not cause a digit to carry. When you record an experimental value, you are supposed to record one more digit than your device can differentiate (ex. guessing the location of a meniscus between two numbers in a graduated cylinder). As such, the final digit that you use in your calculations is inaccurate anyways. Because of this, it is safe to assume that $x$ did not cause any additional overflow. If we were to assume that x overflowed, then it could cause your doubtful digit to also carry, which would result in numbers that disagree with the measured value.

I hope this helps (I know I went into a lot more detail than you were looking for).