The magic number (613) apparently dates from the 3rd century CE, when rabbi Simlai claimed that there are exactly 613 commandments in the Torah. And the rabbis have stuck with that number ever since, because the number 613 is so darned special.
What's so special about it, you ask?
Well, the number 611 is the numerical value (gematria) of the word "Torah" in Hebrew. And the rabbis claim that there are only two commandments (the first two in the Jewish list of the ten commandments) that are directly commanded by God. When you add 611 to 2 you get 613. So 613 must be the correct number of God's commandments.
And in case that isn't enough to convince you, consider this: There are, they say, 365 negative ("Thou shalt not....") commandments -- one for every day of the year; and 248 positive ("Thou shalt...") commandments -- one for every bone in the human body. (The the rabbis were wrong about the number of bones; there are only 206 bones in the human body. But they've tried to fix that by adding the number of major organs to the number of bones to keep the special significance of the number 248.)
Finally, the gematria of each of the four tassels (tzitzit) in the Jewish prayer shawl is equal to 600. Each tassel has eight threads with five knots, for a total of 13. Adding these numbers together makes 613.
So, clearly, we are completely locked in to 613 as the total number of commandments in the Torah.
I must, therefore, have missed 76 commandments. Let me know if you can find any of them and I'll add them to the list.