But the question was.. should I just put a maximum number of bombs the player has to use wisely or should I think something else? At the end I came up with a "bomb bar" which I think is a bit more intuitive. It slowly decreases if you throw bombs too frequently. So you do have infinite bombs, but still you cannot throw them too fast. :)
The bomb bar slowly recovers if no bombs are thrown for a small amount of time, but if you want it to become full in no time you will have to get one of these bomb boxes!
And here's a small screen shot of the first ui: