From seed, it takes 3-5 years to grow to full size, depending on its conditions (availability of food, light, etc.). From plantlet/division or via tissue culture (grown in a test tube with special nutrients), a new flytrap can get to adult size in 1-2 years, sometimes faster.
Small correction to above, "typical" flytraps can get traps that are close to 2 inches maximum size. But it's far more common of a "typical" flytrap to max out around 1 inch trap size, although some will be bigger, especially if the parents had large traps (my friend has a seedling he claims is crossed between B52 and Low Giant; the seedlings are 5 or 6 years old and their traps are 2 inches long).
I once bought a typical from Lowe's whose traps were 1.75 inches long the very first season, and were 2 inches the next season.
Maximum size of flytraps' traps depends primarily on the plant itself. Some flytraps at maximum size only make traps that are 1/2-inch or so in size (e.g. Cupped Trap, Wacky Traps), while others like B52 and DC XL can make traps over 2 inches long.
The largest claimed trap size I've ever heard of was B52 at 2.5 inches, but the largest I've ever seen personally was on a B52 at just over 2 inches.
The DC XL is a newer cultivar only sold here at FlytrapStore and it develops traps that often outmeasure same-aged B52's.
There is no evidence that I know of to prove that a flytrap that never eats bugs in its life will have smaller traps at adulthood than a genetically-identical flytrap that eats insects regularly (e.g. take two B52 plantlets and "starve" one but feed the other). There IS proof that a flytrap that eats regularly will grow faster than a flytrap that does not eat bugs. Flytraps that never eats bugs will not starve, unless they are already getting pitifully-inadequate amounts of sunlight.
So to get your flytrap as big as possible, simply keep it alive.
If you want a flytrap with gigantic traps, and if whatever flytrap you have isn't cutting it, you might consider purchasing a DC XL, B52, A2, King Henry, or Low Giant. Of the flytraps I am familiar with, those five produce the absolute largest traps, at around 2 inches/5cm, plus or minus about 1cm for the first two (DC XL, B52). The latter three probably won't produce any traps over 2 inches or just above two inches; at least not that I've ever seen nor heard of.
Low Giant is going to be the easiest to get a hold of out of that bunch, followed by B52. A2 and DC XL are very hard to come by, and only FlytrapCare sells them as far as I know, but are both sold out atm. I don't know anywhere offhand that might be selling a King Henry.