Are 3-8 input NAND gates common?
Try searching yourself at a distributor such as Digikey to get some idea of availability. 

I'll take you through a step-by-step if I was looking for, say, a NOR gate with at least 4 inputs. I'll be primarily interested in 4000 series CMOS or 74HC CMOS families (usually it would not be for the same application). 

First, get on the site and search for NOR gates under "gates and inverters" and select "In Stock". 



Now highlight the number of inputs as shown above, and click "Apply Filters". 



Then narrow it down to the logic families of interest as shown above and hit "Apply Filters". 



Then drill down to reasonable packaging (assuming you're not going to buy a reel). You know what to do.. 



Only one page remains, now sort it by decreasing available quantity by clicking the column head, as above. The sorted column will be highlighted (grayed). 

You can see that about the only choice available with the above constraints is various versions of the 74HC4002 or the 4002, and they seem to be fairly available. It's a dual 4-input gate.  

You can repeat this for whatever gate you are looking for, and it takes very little time. 

You can cascade gates, however you obviously lose one gate's worth of propagation time every time you go through another gate in series. If you combine two 4-input gates with a 2-input gate to get an 8-input gate, you will have two propagation delays.