Analog circuit to turn on LED for 10 seconds just after power is applied?
Is there any way to do this with just a capacitor/resistor combination instead of an IC?

A: Capacitor from +5V to resistor to LED to ground.
 Reverse diode across LED. 

When 5V steps o -> +5 LED will light.

I_LED initial for red LED ~= (Vcc-Vled)/R = (5-2)/R = 3/R for red LED
 Or R = 3/I

LED will dim exponentially.

Time constant ~~~= RC.
 So C =~ t/R

For say 20 MA initial R = 3/I = 3/0.020 = 150 ohm.
 C = t/R = 10/150 =~ 0.06 Farad = 60,000 uF.
 That's doable with normal caps but large.
 An appropriately rated supercap would be easier to do.

Doing this with a single super cheap transistor and a few Rs and cs is so easy and so superior in result that the cap and R solution makes zero sense in almost any context.



Supercap vs Standard Al electrolytic - not as big a difference as many would assume:

The phantom commentless downvoter is back.
 IF this is for the statement that   


"that's doable with normal caps but large"   


then it shows a lack of understanding of what was being said
 AND a lack of appreciation of what is available. 

A supercap would probably be the best choice but the point was (as the words say) that you COULD do this with conventional caps BUT it is at the very high end of the range where you would. 

You can buy an eg a 47 mF 5V5 supercap here  for $1.58/1
 and you can buy a 47 mF 10V standard aluminium electrolytic cap here for $3.75/1. In this case the supercap is about 42% of the price of the Al cap BUT the Al cap is 10V rated so can have = 3.3 x as much energy at full charge so in terms of energy storage per $ the std cap is cheaper. ie a bit of looking around would probably find a lower voltage Al std cap that is cheaper for the supercap wity the same desired spec. BUT the differebnces are close enough that it's not really important in most cases. Other attributes would make the difference.
 eg  the std cap is much much much less susceptible to over voltage damage,
 BUT it's far larger - in many cases the supercap would be the preferred choice. But not all.  

"Horses for courses", and be wary of forming opinions about what "old tech" is capable of just because "new tech" seems to be so much better. Often it may be. Sometimes it isn't.