I think as with everything u/s should be used with caution and kept to a minimum so I would ask at every stage whether a u/s is really necessary or whether the same info can be obtained the via a different method.
There's a fine line between the benefits (which are clear) and any potential risks, and only you can decide where that line lies. I've refused a u/s to check baby's position (at 28 weeks) and requested that a MW with a lot more experience come and prod me to see whether the position could be determined. It could, which personally I judged less risky than a u/s, even allowing for potential error. I wouldn't necessarily make the same call at a 36 week check because a u/s gives additional information but by their own admission they were planning on scanning me purely for foetal orientation and not because they had concerns about the placenta, the cord, whether there was sufficient room or anything else - just because baby wasn't obviously head down, which isn't a reliable inducator at that stage anyway. They were right, it wasn't head down, and it's probably better to be aware of that than not but a u/s wasn't necessary.
As with everything a healthy aversion to risk is a good thing so you're not being paranoid. Too much, especially prolonged exposure, could, in theory, be harmful. But noone is going to subject a pregnant woman to that level of risk to find out, which means that in all likelihood you're fine :)
I'd had 5 as well by that point - 2 early and 3 attempts at a NT measurement and I'm up to 8 with definitely 1, probably 2 to go.