156036

First, I converted all the fractions to the common denominator. I did it by finding the LCM of all the denominators which resulted in the common denominator of 600.

My fractions were as follows:
1/6 = 100/600
1/25 = 24/600
3/5 = 360/600
3/20 = 90/600
4/15 = 160/600
5/8 = 375/600

I then was able to add the fractions and in order to get closes to one I needed to get the numerator as close to 600 as possible.

After trial and error I concluded that the closest I can get to 1, was to use the following fractions:
375/600 + 100/600 + 24/600 + 90/600 = 589/600

So, the original fractions I used were: 5/8 + 1/6 + 1/25 + 3/20