No it is not 1.
The 3’s are infinite.
.3 is 3 Tenths
.33 is 33 Hundredths
.333 is 333 Thousandths
.3333 is 3,333 Ten Thousandths
.33333 is 33,333 Hundred Thousandths
And on and on.
All you are doing is simply going to the next decimal position and placing a 3 in that position, and you do that infinitely.
.333… x 3 = .999…
9 Tenths, and you keep on adding 9’s in the next decimal position, which there are infinite decimal positions.
You are working to the right of the decimal point. First you place a 9 in the Tenths position. Then you place a 9 in the Hundredths position. Then a 9 in the Thousandths position.
So you have .999 at that point, and you keep placing a new 9 in the next decimal place to the right of the previously filled position.
The result is 9 Tenths and some change. That’s all that needs to be said, because the 9 Tenths will never change to a 1.0. Never! You are only adding another 9 to the end of the string. The 9 in the Tenths position never changes, it is FINAL!
9 Tenths and some change is less than 10 Tenths!
9 is less than 10
99 is less than 100
999 is less than 1,000
9,999 is less than 10,000
99,999 is less than 100,000
and on and on!
Do the same to the right side of the decimal point:
.9 is less than 1.0
.99 is less than 1.0
.999 is less than 1.0
.9999 is less than 1.0
.99999 is less than 1.0
…
.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 is less than 1.0
…
.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 is less than 1.0
No matter how many 9’s you place after the decimal point it will ALWAYS be less than 1.0! Period! 
It’s not like you are adding to an existing decimal place, as if .9 + .1 = 1.0. You are not adding anything to the existing 9’s. You are adding an additional decimal place with a 9 in it. Never does adding a new decimal position at the end of the previous string of 9’s turn the total to 1.0. Never!