The biggest cost — assuming you don't crash — is to your productivity. In part, that's a simple consequence of splitting your attention, so that you're partially engaged in multiple activities but rarely fully engaged in any one. In part, it's because when you switch away from a primary task to do something else, you're increasing the time it takes to finish that task by an average of 25 per cent.Not only that, but the constant switching from task to task burns up your limited resources of energy and attention. To use a computer analogy: you can write a program that fetches (retrieves) one row at a time from a database, or you can write the same program to do a single bulk fetch of many rows. The single-row approach is much less effective, because of the overhead of switching back and forth between your program and the database engine. The bulk fetch is more effective by far, because it eliminates most of the overhead.
In the same way, when we multitask we spend a lot of our energy on simply switching back and forth between the several things we've got going. When we dedicate ourselves to a single task at a time, we still spend the same amount of time doing the tasks. But we save time overall because we have spend less time managing the tasks.