How many splices can a junction box contain?
As many as the box fill will let you have

The limit on box fill is set in term of the number of wires entering the box -- splices don't count against it.  That said, it's time to jump right into the box fill calculations.

You have 9 10AWG box fill units there (1 for each hot and neutral entering, and 1 for all the grounds taken together).  A quick glance at NEC Table 314.16(A) says that a 100x54mm (4" by 2.125" deep) square box has room for 12 10AWG fill units in it, so you're good to go.  If there was an internal clamp, this raises your fill to 10 units (the clamps taken together count as 1 unit as per 314.16(B)(2)), but that's still within the 12-unit limit for 10AWG.

Note that this simplified method relies on all the wires coming into the box being the same size.  If you have different sizes of wire involved, you'll have to use the cubic-inches method with Table 314.16(B).

One last thing -- perhaps you could use a magnet on the end of some fishtape to fish the screw out of the conduit?