You can get as many as you want, provided you follow the immigration and naturalisation laws in the country of which you wish to become a citizen. It's not easy, but it's possible.

For instance, I am entitled to Israeli citizenship but have zero interest in claiming it.

I seem to remember we discussed this a few months back. Provided that all four of your grandparents are citizens of different countries, all of which allow them to pass citizenship along to their grandchildren (e.g., Ireland, Italy) and that both of your parents are citizens of two further, different countries by virtue of their birth, both of which allow them to pass citizenship along to their children, and provided that you are born in a country different from all of these that allows you to claim citizenship by virture of having been born there, then I would imagine the maximum number of citizenships you could be born with is seven. I cannot see how it could practically be any more than this.