Adding C-wire to my system

You have a common wire (COM 24v "C") blue wire on your terminal strip.

But I question if the transformer in the board is large enough to power a Nest.

Is that a transformer on the right above the circuit board?

If not you may have to use a separate transformer and relay. Nest's need a fair amount of power to charge their battery.

@EdTheHeaterman will comment

See @EBEBRATT-Ed 's comment above — and I suspect @EdTheHeaterMan will chime in. But no, just because it's smaller the new Nest still uses a lot of power.

At the thermostat you have G, R, Y, and W connected using the appropriate colors: Green Red Yellow and White.  The furnace has no yellow wire connected.  but that can still operate an air conditioner adequately.  It may actually work better if you connect the Y on the furnace circuit board, but maybe it is ok without that connection. At the furnace there is a blue wire connected to the C terminal.  I suspect that is from the outdoor condenser.  This assumes that you have air conditioning attached to that furnace.  

The wire at the furnace does not look contiguous (the same wire from end to end)  the Green at the thermostat is a much darker shade of Green than the Green at the furnace.   Is there a chance that the furnace wire is connected to the outdoor unit and the thermostat wire at a location outside the furnace? 

Is there an unused wire from the thermostat to the "out of furnace" junction area?  If yes, then it is easy to add the C wire to the thermostat and to add the Y to the furnace also.  

I am suggesting that you connect the Y on the furnace to the Y from the thermostat because in many cases, the Y on the furnace will activate onboard software logic to let the furnace know that the G is being activated for Cooling and not just for Fan On operation. This feature may enable some fan time delay, or operate a different fan speed, or both, when there is a call for cooling. Those features are not needed if there is no call for cooling and you just want to switch the furnace fan from Fan Auto to Fan On at the thermostat.

As far as the comment about the NEST using too much power for that furnace transformer, that is usually not an issue. That transformer is designed to be used to operate a wide range of cooling system relays, and also humidifiers for heating system usage. I have rarely had a problem connecting the NEST to a furnace transformer.

Sometimes it is difficult for some folks to follow wiring diagrams so I thought I would post the before and after diagram. Of course, I don't have a crystal ball so the BEFORE is a guess at how the wires are connected, So @domolek will need to verify the "Before" diagram.

I have seen the “Before” wiring several times in my short 45 years of doing this stuff.  As long as there is an extra unused wire in the line from the thermostat to the junction area, then it should be easy to do the "After" diagram.  I don't know what color the extra wire might be on the line from the thermostat to the junction area, so I selected light blue to indicate that wire.  But you can use whatever color the extra wire is, because electric is colorblind. 

Hope this makes this idea as clear as can be