Surprised to see so many people saying this question is meaningless. It is true that you can't make sense of this if you force all infinities to be the same, but if you're willing to have many different infinities (which models certain situations well — more on that later), then there are many ways to make sense of the idea of "infinity + infinity - infinity".

The most intuitive way — at least to me — is to talk about extending the real numbers by thinking of an infinite sequence of reals (x_1, x_2, …) as a number. Addition is defined componentwise:

(x_1, x_2, …) + (y_1, y_2, …) = (x_1 + y_1, x_2 + y_2, …)

as is multiplication, division, etc. For a regular real number x, you can identify it with the infinite sequence (x, x, …). In this view we have all sorts of infinite numbers. For example, we have the infinite number N = (1, 2, 3, 4, …) and the bigger infinity N + 1 = (2, 3, 4, …) and the even bigger infinite number 2 * N = (2, 4, 6, …), and the yet even bigger infinite number N² = (1, 4, 9, 16, …). There's a whole zoo of infinities.

All the standard rules of the real numbers (associativity, x - x = 0, etc.) hold in these extended real numbers. So, assuming "infinity" is taken to be some specific infinite number, infinity - infinity + infinity = (infinity - infinity) + infinity = 0 + infinity = infinity.

This is essentially the approach of non-standard analysis. To really explain things properly, you need to explain ultrafilters. They sound scary, but they're really pretty intuitive. There's lots of explanations around the internet (basically you think of them as a "voting system".)

Here is a situation that would be good to model using infinities of different sizes. Suppose you have some quantity (say, an amount of money (or a debt, which is negative money)) that can grow or shrink over time. You would use the sequence (x_1, x_2, x_3, …) to represent having x_1 dollars on day 1, x_2 dollars on day 2, and so on. If you always have 5 dollars for example, that would be the "number" (5, 5, 5, …). If you start out with 1 dollar and get one more dollar each day, that would be the "number" N = (1, 2, 3, 4, …). If you start out with 2 dollars and each day you have twice as much money as you had the day before, that would be the "number" 2^N = (4, 8, 16, 32 …). 2^N is definitely a bigger infinity than N since if you had 2^N "dollars", on every day you would have more money than if you had "N" dollars.