I would like to use the most discrete parts possible.
I have a solution which requires 7 555's, 20 transistors, and about 100 resistors. Do you need to use more than that?
Seriously, I firstly assumed you meant that you wanted to use mostly discrete components, but then you mentioned the 555. Do you mean that you want to use commonly available, through-hole components maybe?
Kris has alluded to some problems, those of power output and efficiency, but let me add another one or two into the mix.
There are two major issues that I see.
The first, and the major one, is that the phone will expect a charger which can supply a certain current. If you cannot maintain that current, the phone may decide to refuse to charge the battery. If that current is 500mA, then you require 2.5W, and you're simply not going to get that from the panel. There are two ways to resolve this problem. The first is to get a much larger solar panel -- say, a 5W panel. This will work as long as it is in full sun for maybe 6 hours per day (more in the tropics, less at higher latitudes).
If you can't manage a larger panel, then your option is to use the panel to trickle charge a larger battery, and to use this battery to provide the power to charge the phone. This solution would be useful even with a larger panel because it would allow the phone to charge at night or when the sun was obscured by clouds (presuming that the battery had been charged earlier.
The second issue may be important if you need the highest efficiency, that is of obtaining the most power possible from the panel. An MPPT (Maximum Power Point Tracking regulator will provide a load to the solar panel that varies such that the panel transfers the most power. This is commonly used in circuits used for charging batteries, so it really requires that you have a battery to charge (not a load to power).
An apparently simpler solution might be to take the battery out of the phone and charge it externally. This would be probably the most efficient way to do it, but you have no phone during this time (unless you swap batteries) and you must be VERY careful not to overcharge the batteries -- this will kill them really fast, sometimes explosively (and I'm not kidding).
If I were doing this and I had a phone with an easily removable battery I would go the battery swap method. If the battery was not easily removable (as in my case it is not) I would charge an external battery (say a 12V lead acid battery) and then use this to connect to my phone's charger port via a switchmode regulator.
So what does all of this mean? Basically you need to determine:
- Whether this is practical from an energy vs. time standpoint
- How you're going to charge the phone
- What modules you will require.
- (and finally) how you can build this.
Beware that most MPPT devices use a microcontroller. MPPT probably adds about 20% to the energy you can get form a solar panel (more if the panel and the battery are poorly matched). If you can live without MPPT then your design process will be easier. The good news is that with an appropriate solar panel, MPPT can be added later to boost your performance.