Screenshots:
http://imgur.com/bTh3KTV
http://imgur.com/hZTdEZd


I don't think I added any tariff, at least not on purpose. On the dashboard on the right where it lists my name, number, credit, etc it says this:

Aktueller Tarif: 9 Cent Einheitstarif

Tarif-Optionen:
Smart- & Allnet-Optionen: buchbar
Internet-Optionen: buchbar
Sprach- & SMS-Optionen: buchbar
Auslands-Optionen: buchbar
blau Kostenschutz: aktiv

The 100MB and 1GB options are both available (they cost less than the €14.81 credit the desktop site thinks I have). It's only the more expensive 3GB and 5GB plans that say I can't add them because my credit is low ("Ihr Guthaben reicht für die Optionsbuchung nicht aus").

Is there a code I can text to Blau to add the 5GB package? Maybe that will work even though the site thinks my credit is low.

That looks good (in the sense that there's no active tariff option), except for your balance on the customer portal which obviously lags behind. You should wait a couple of hours and retry ordering through the app or their website and if both fails your only option left is ordering your bundle through the voice-guided menu available at 1155, however I'm not sure if calls to this number are being charged when calling from the US. So better wait until you step on German soil before you burn your credit quickly. Depending on how well your German is, you may also find it difficult to cope with the spoken instructions. I was able to find a two years old forum post where someone described the key sequence for ordering the 5GB data pack:
call 1155, press "8", "1", "1", "2"
However I'm note sure if that is still valid as the tariff portofolio has seen changes since then.

Thanks a lot, I appreciate all the legwork! I only know beginner's German but I should be able to pick out the words for internet and five so hopefully I'll manage. If all else fails, I'll order the 1GB package and top it up for €3 at a time which should get me 4GB for under €20 which is fine by me. I expect to use about 3-4GB.

Better don't do that! If you order the 1GB bundle you cannot order any other data bundle for the remaining billing period of 30 days. During that time you can only reset your data counter for € 3 (so-called Highspeed-Reset). This, by the way, can be repeated multiple times, however these resets do not extend validity of your data allowance.

I think that is okay for me. I need 3-4GB for about 25 days and then I will be leaving Germany. So I was going to do the following:

1. Buy 1GB — €9.90
2. Add another 1GB when I run out — €3
3. Add another 1GB when I run out — €3
4. Add another 1GB when I run out — €3

Total: 4GB for €18.90, for 30 days. Is this correct, or am I missing something?

True, it is more expensive than the 5GB package, but it accomplishes what I want and the price is reasonable to me. I will wait until I arrive in Germany (in about 12 hours) and check the web site one last time.

That's correct. You can do it like this. Have a good flight!