I'm just blasting this out, so it's kind of long (sorry, no time to edit). An easy way to look at this problem is in terms of energy (e.g., joules or watt-hours). Since the r-pi consumes about 1 watts of power (i.e., 1 watt-hour of energy per hour), let's use watt-hours. Running an r-pi (and nothing else) on a 24x7 basis would thus require 24 watt-hours per day, and 168 watt-hours per week.
So, if you want an r-pi to run on batteries alone for 1 day, you would need 24 watt-hour battery pack. If you were to obtain 10 rechargeable AA NiMH cells (such as the Energizer or Durcell 2500 mAH batteries available at most grocery stores), and put them in series, you'd have a 12-volt lead-free power source (rechargeable AA batteries are 1.2V rather than 1.5V). The total energy in such a pack would be 2.5 AH (2500 mAH) x 12 V = 30 watt-hours. If fully charged, this would be sufficient to run an r-pi for 24 hours (24 watt-hours of energy) without totally killing the battery pack (20% reserve). To extend this to a week you would obviously need 7 such packs.
Now, for the solar part. To operate continuously with a solar panel only, you need a panel that can deliver AT LEAST 24 watt-hours every day, all day, and the sun must sufficiently direct and bright to deliver the necessary light energy ("insolation") that was used as a reference for the solar panels specification. This is where the problems start. First, it's tough to find a terrestrial location where the sun shines brightly on a 24x7 basis, with no clouds, etc. Even if we did, most panels assume insolation levels of 1000 watts per square meter in their specs, per lab conditions, but in reality, it's rare to get even 80% of that due to atmospheric interference, time of day, etc. However, for the sake of argument, and because I like round numbers, let's assume a panel that claims 10 watts of output in bright sun will actually deliver 10 watts.
Brief but relevant digression: I live in Seattle. According to aclassic song, "The bluest skies you've ever seen are in Seattle." According to a post card I once saw, entitled "Seasons in Seattle," the four seasons were simply depicted as four shades of gray. From this you can glean that if you pick just the right time to look at the sky, it's a very blue sky. On average, however, it is apparently some shade of gray. My hypothesis is that the frequent-but-usually-light-rain keeps the sky so smog-free that you can sometimes catch a glimpse of blue sky.
The reason this is relevant is that insolation levels for most locations on earth are available either as statistical measurements or predictions. If I recall correctly, the annual AVERAGE daily insolation in Seattle is on the order of 2 hours per day, whereas in a smog-free area of the southwest USA, it might be 6 hours per day (I might be remembering seasonal averages though). That's good enough for back-of-the-envelope calculations, to make my point. If you need 24 watt-hours of energy to run an r-pi, and you want to use solar to power it in Seattle, then you'll need sufficient PV capacitity to collect all of that daily energy (plus overhead due to inefficiencies) within a 2-hour window, on average. Thus, in a perfect world, you would need at least a 12-watt solar panel (24 watt-hours / 2 hours = 12 watts). The problem is that it's not a perfect world, and 2 hours of insolation is only an average, and there are many inefficiencies.
There are periods of abundant sunlight, and long periods with none at all, so many off-grid PV-solar-powered systems typically provide perhaps a week of ride-through, which is why I mentioned earlier that it would take 7 10-cell battery packs to last for 7 days. At about $10 for a set of four such batteries, that's about $25 for 10 of them (1 day's worth), and $175 for 70 of them (a week's worth). Of course, that's with consumer NiMH batteries. If you don't mind SLA batteries (sealed lead-acid), you could just get a pretty big deep-cycle battery for under $100, and end up with 3X the capacity (which is good, because you can't fully deep-cycle them very often, or they die).
Alternatively, because our gear gets sealed in radio-tight and air-tight containers (where SLA batteries cannot be used), I rather like the industrial Lithium Polymer power packs from http://Tekkeon.com
. Each has a built in charger and regulator, and can deliver 58 watt-hours at a range of voltages between 5V and 19V (DIP-switch-settable), at currents up to 4A. There's also an aux USB power port that operates in parallel. It accepts many types of charging power, including solar, vehicle DC, and AC. Multiple packs can be daisy-chained.
When sunlight is abundant, a larger battery capacity provides a place to store it, and when it's not, the larger battery capacity provides ride-through until the sunlight returns.
I like to seriously de-rate off-grid systems, since their usual focus is often related to survivability. Use the insolation available for your area (search Google for an insolation map, etc.), assume no more than 80% will be available at the PV panel, then account for the inefficiencies. PV panels only output their rated values when directly pointed at the sun, and less otherwise. The open-circuit output of a PV panel will be dragged down by the batteries (which present a load on the panel), so a voltage which is sufficient to charge empty batteries will no longer be adequate to top them off. To solve that problem, and extract maximum power from a PV panel, you need a maximum power point tracking (MPPT) circuit. An MPPT circuit is essentially an any-to-any DC-to-DC voltage and current converter, which essentially converts whatever power the PV panel can provide to the optimal combination of voltage and current that is needed (such as for charging). While MPPT chips are available for a couple of bucks, the smallest MPPT converter I've ever seen was on the order of $250 for one that would handle a modest 60 watts (e.g., up to 3 20-watt panels). This is 10X the proposed r-pi price, and it doesn't even include a PV panel.