I tried this a few times on AskReddit, but only got "porn" and "browse reddit" as answers. I still think it's an interesting question, so I'll try here.
Details
1. You will enter the room with the computer exactly 14 days from now. This is to allow you to prepare your plan and your software/extra hardware. 2. You are locked inside the computer room for all 24 hours. You bring all your stuff, come in and the door is locked. Food, beverages and bathroom are provided. (This computer is too big to move anyway). 3. Absolutely no software is provided with the computer. So bring your own OS. 4. So you know what software you can run, we'll say it's x86_64. 5. The computer comes with 4 USB ports, one HDMI, one VGA, one multi-card reader, one ethernet port, a monitor, and a keyboard and mouse. That plus the computer itself is all you're given. 6. Everything inside this computer is infinitely fast. So there's no waiting for CPU, GPU (which isn't all that necessary anyway since you have an infinitely fast CPU), read/write to RAM or hard disk, or the bus. 7. However, your internet connection, since it relies on other computers, is limited to Google Fiber speeds. 8. This is the only connection that is provided. You can, however, bring other network hardware if you want. 9. You are also limited in the amount of RAM and storage space you have, even though you can write to them infinitely fast. You get 16 GB RAM and a 1 TB hard drive. 10. You can add more RAM or disk space, but that would be your own hardware that you would have to bring, and it would operate at normal speed. 11. You have cell reception, in case you wanted to bring your own modem for that for some reason. 12. I've already mentioned bringing some of your own hardware, but I'll extend that to say you're allowed to bring absolutely anything you want into the room, as long as it'll fit. The room is 20 m x 20 m (and 5 m tall) of usable space with the computer in one corner.
You may now begin.
"Genetic algorithms" was the first thing I could think of, since they take time on classical computers and can often produce results that human designers aren't expecting. Furthermore, they don't require any significant storage space, since they don't need to remember anything about previous generations, so your 16 GB limit is not particularly restrictive. As a summary, and skip the next three paragraphs if you already know how a GA works: they mimic biological evolution by randomly recombining the "genome" of anything you want to optimize, such as a telephone network, chip design, road-map for a proposed highway system, and so-on. You begin with two randomly generated "parents" and, on each loop, recombine randomly chosen "genes" from each parent to produce five offspring. What you then need is some kind of scoring algorithm, which could be a simple formula or an entire simulation (estimated traffic of an airline network for a year, for example), which can calculate a score for each offspring. You pick the two best scoring children and pass them on to the next generation. Lather, rinse, repeat. You can vary this by tracking multiple "strains" and allowing them to cross-breed, and you can add the element of mutation by randomly changing one of the genes every X-many generations. Genetic Algorithms have been used to optimize modern telephone networks (cost, capacity, damage resilience), power efficient chips (I understand GAs are being used to optimize the ARM architecture these days), and one interesting case was the first known application: circuits that can distinguish between two musical notes. The result from a few thousand generations was a circuit--programmed on a FPGA--that had a totally unconnected sub-circuit. According to electrical theory, this sub-circuit could not serve any function, but on the physical FPGA chip was acting like a kind of resonance chamber, like the hole in a guitar, inducting a small current that in turn influenced the rest of the circuit. Such an invention was completely unexpected by the programmer, who had no idea such a circuit was even possible. You run a GA as long as you have the budget or patience for. Sometimes you get excellent results after a weekend, but the longer you run it the closer it asymptotically reaches the ideal implied by your scoring algorithm. You might stop it once it looks like you've found the "ceiling" in the scoring criteria itself. So I'd bring any reasonable OS and a program that runs a Genetic Algorithm that would essentially do the equivalent of asking a Genie for "infinite wishes": a design for a computer that would also have infinite processing speed, or at least the best possible. Might also run a GA to get the best performance-per-watt design for mobile computers. Some other ideas, perhaps loaded up to run in series, although they would theoretically all complete the instant I hit "Enter": A molecule with the lowest mass, highest strength (sheer and tensile), highest melting point and lowest cost that the best simulation of physical materials (my scoring algorithm) can judge. I'll come out and start a Space Elevator company. A language translation matrix that can satisfy the largest and most varied collection of human-authored and curated "Rosetta Stones" that I can fit on any media I can bring into the room, both text and spoken, including dialects, slang, and inflection. Might solve world peace if we knew what each other was really saying. Solve every conceivable "Traveling Salesman" route for every mail carrier and delivery company that I can sign-up in my 14-day prelude. Ka-ching. This might actually be the one that takes the longest to run, because I'll be filling up a lot of conventional Flash/portable hard drives with the results. Find the safest, cheapest and densest way to store hydrogen that my physics simulator can judge, then sell the answer to automobile and fuel-cell makers. Same for "supercapacitors" and sell it to electric car makers. Solve Chess. The potential of GAs starts to become a bit "magic wand-y" after you think about them for a while. Their limitation, assuming an infinitely fast computer, is really the scoring algorithm, and their limitation is human imagination itself.
Since it's infinitely fast, even uncountably infinite spaces could be iterated through instantly, so I would have it test every one of the uncountably infinite number of possible proofs that P = NP (and every other unsolved problem) and check all of them for correctness, solving all currently open math problems. Then I'd do what forthewar said. That's all I can think of for now.
Solve Chess. I'd ask the computer to analyse every possible position and determine if White always wins or Black al;ways draws or if the game is only determined by certain openings. I'd also ask to see how many digits of Pi it could muster up. With infinite speed and internet I'd have it search the entire www and formulate its own version of AI. 24 hours at gigabit speed could analyze the entire www. I'd have all the unsolved math problems that I could research with me (some of them have Million dollar prizes) I'd take along the uncracked codes of fame as well. At this point we have only looked at the chapter headings of DNA. I would like to unravel the "Book" that appears inside the cells and gain the trapped knowledge residing in each of us. But this is ridiculous, what infinitely fast computer uses OS of today? I think the limits in DETAILS are just meaningless, I would have the computer WRITE new software that would run at its own rate. With infinite speed it could run through all possibilities and write its own software with the twist of a wrist. I'd map out evry possible universe and every possible situation and run the results to determine what will be the outcome in OUR universe as well as all the other ones possible. That would leave me with a whole bunch of time, so then I would get wild with it and remain as creative as I could trying to gain as much for humankind as I could. I'd upload to cloud, who needs hard drives. I'd interconnect with other computers, 16GB ram haha. I would do the whole world the ultimate favor and by the next day the world would have changed forever.
Here is what I would bring with me: 1. If I can have any OS I want, I'll bring Win 7. If I actually have to have a copy of the OS, I'll bring Ubuntu. 2. At least one other monitor. 3. At least one external HD. Once that has been taken care of, I would install BOINC and start up as many projects as I can. This computer could probably make a pretty big dent in their data processing. While that is flying away on one screen, I would probably spend the rest of my time gaming. If that ever got boring I would watch YouTube videos of Corgis. Finally I would download as many books as I can from Project Gutenberg and other similar sites and store them on my HD(s). That will provide a good amount of entertainment for a while.
I think I would install linux (duh) and see if I can crack a couple of really strong encrypted stuff. US army maybe? Just as a proof that this is possible and if I can have access to such a powerful computer, someone else probably has as well. And with an infinitely fast computer, encryption fades as snow melts in the sun. I could also try to run some simulations on some complex engineering problems I might be working on at the time. That should be fun :P Just for completeness, not all of us are US citizens. What are the google fiber speeds? I'm assuming 100Mbit/s here.
To mine fast enough to make more than you could get from interest in a savings account you'll be spending more on electricity than you make. That's by design, and it's hardly a secret. If bitcoins grew on trees they wouldn't be worth anything. I'm not sold on bitcoin either, but if crypto-currency is workable at all it can't be possible to easily generate as many as you want. Given an oracle machine for generating bitcoins, you wouldn't have that problem. You'd get however many of the ~21million maximum bitcoins that hadn't been mined yet, and either be very rich in currency you can't easily spend or kill the bitcoin economy, depending on how much you've caused the exchange rate to fall and how the bitcoin ecosystem reacts to the trickle of new coins stopping.