Without being able to see the cables as they enter the cabinet; or the ability to touch or trace them, here is what I assume is going on.
Definitions:
Grounded (neutral) from the service
A typical single split phase service is made up of 3 wires. Two ungrounded (hot) conductors, and one grounded (neutral) conductor. The ungrounded (hot) conductors will connect to the main service panel through a disconnect (usually a large breaker), while the grounded (neutral) connects to the neutral lug. The neutral lug will be bonded (electrically connected) to the neutral bus bar, and all grounded (neutral) branch circuit conductors will terminate at the neutral bus.
Grounding Electrode Conductor
This conductor is used to connect the grounding electrode (ground rod, etc.), to the grounding bus in the panel. All equipment grounding conductors will be connected to this bus.
Bonding Jumper
The bonding jumper is used to bond (electrically connect), the un-energized metal parts of the panel to the grounding system.
Assumption:
Since it appears that (what I assume is) the grounding electrode conductor terminates at the neutral bus, I'm also assuming that this is the main service disconnect. This leads me to believe that the neutral and grounding buses are bonded (electrically connected). In which case, technically, grounded (neutral) branch circuit conductors can terminate at the grounding bus.
So you have two options:
Terminate the grounded (neutral) from the new circuit to the grounding bus.
Move the green wire that is terminated on the neutral bus, to the grounding bus. Then terminate the grounded (neutral) from the new circuit, to the freed up slot on the neutral bus.
Additional Information and Code Compliance:
Number of Conductors
Since this is a new circuit, it has to be installed to current code standards.
National Electrical Code 2011
ARTICLE 250 — GROUNDING AND BONDING
VI. Equipment Grounding and Equipment Grounding Conductors
250.140 Frames of Ranges and Clothes Dryers. Frames of electric ranges, wall-mounted ovens, counter-mounted cooking units, clothes dryers, and outlet or junction boxes that are part of the circuit for these appliances shall be connected to the equipment grounding conductor in the manner specified by 250.134 or 250.138.
Which in this case means installing a NEMA 14 receptacle for the dryer, and a proper grounding conductor.
You'll have to follow the dryer manufacturers installation instructions for upgrading to a 4 wire cord. For more information see this answer, and this answer.
Since you've said that you're already using 4 wire cable, you'll simply have to terminate the grounding conductor in the cable to the grounding bus in the service panel. Then connect the other end of the grounding conductor to the grounding terminal in the dryer receptacle.
Size of Conductors
You'll also want to be sure that you're using the proper size breaker and conductors. In the case of a dryer, you'll typically use a 30 ampere breaker and 10 AWG conductors (depending on the length of the run). However, you'll want to check the dryer manufacturers installation instructions to verify this.
Best Answer
As other comments have stated, it is likely an AFCI or (more likely) a GFCI breaker. Can you determine where that circuit goes? Knowing its termination would help, since some areas, like bathrooms and laundry rooms, are more likely to validate that. Also, it's hard to tell from the picture, but the two white wires shouldn't be connected to each other at the screw terminal. If they are connected, I'm at a loss for what it could be, but if they aren't, then it's some sort of fault protection breaker. I'm a bit surprised by the lack of a test button, though.
It doesn't appear that you have room for a 2-pole breaker to provide 240V for an EV charger. You would need to see the blades on both busbars to indicate room for a 2-pole breaker. In the below image, the red circle shows a blade for the left leg busbar. This is visible in your panel. You would also need the blue circle (right leg busbar) to be exposed.
I know you didn't ask about this, but it seems like it might be a good time for a panel upgrade. You've got five 15A breakers, two 50A breakers, and five 20A breakers for branch circuits all on (what appear to be) tandem circuit breakers, but only a 100A main breaker, which is even taking up space because it's on the busbars directly instead of connected to a lug.
You might want to consider checking if you can upgrade to a larger panel, physically and electrically. Especially for EV charging, which can be a 30A or greater continuous load, you'll probably want to see about 160A service or higher. Depending on the size of the aluminum service entry wires (I can't see what size they are from the picture), you might not even need to replace those.
Something like a 16-slot, 160A (minimum on both) main breaker panel would help you future-proof your panel a bit, especially for adding EV charging capacity. Also, I was able to find a 20-slot, 200A panel with included circuit and main breakers for $150, so it won't likely cost you too much either.