I wonder if someone could build a new digital currency that starts let's say at 1 trillion units, and is expanded by 2 percent per year (that 2 percent being split so a slight increase happens every day, so it doesn't happen all at once, but after a year, the increase rounds up to 2 percent).

The problem would be seeding it and distributing it early on. But it could still be mined much like Bitcoin, with the difference being that the total amount will never be limited. Also, it should be easier to mine them, since the (non-finite) total (1 trillion) would be a lot larger than the 21 limit for Bitcoins. Someone should also do the math for this new currency's adoption rate, so all of the 1 trillion units (adjusted with the 2 percent inflation rate) get distributed in 3-5 decades.

Basically we don't want a few thousand people to get that whole trillion within 10 years. I'd also prefer if anonymity would be built it (look to Zerocoin for that). Perhaps it shouldn't be anonymous by default, but you should be able to "switch it on"). It should also be as easy to use and as secure as possible (the clients), so normal people can use it safely on all of their devices.