It really varies depending on what you mean with "getting to this power within one week".
If you mean the Gaining Power event then do note that you get double the pts for buildings, and still it would take a rather alarming amount of time boosts, stacked upon several bonuses of +construction.
If you mean someone registered an account 1 week ago and is now 350M power, then that is a joke.
Not because it's not possible - but because he made the (in my opinion) huge mistake of buying several pure-troop packs.
This will only make his kingdom lose the coming KvKs. Buying T5s is a pure mistake.
For every T5 troop you get roughly 100 power, and last week I had the option (it varies from one player to another, but just to give you a ballpark) to buy a troop pack with 6x75k = 450k T5 troops.
Tule Fööniks said:
some people just gather tons of gold on their account when lvl 5, sitting there only gathering golfd, then relocate to new kingdom to actually build it up with gold reserves
Bullocks. It varies between buildings, but relation between gold to instantly build something and the power it gives moves around the 1:1, more for the compulsory buildings. 350M gold.... Gathering at 1.5min-per-gold (let's be generous with the GGMs) and 3 marches... we're looking at a solid 2 gold per minute = 175M minutes.
And that's the equivalent of the modest amount of 3,329 (and a half) years.