If you have googled this a bit more, you would have found the answer right away.

The box-shadow property syntax is the fallowing :

box-shadow : horizontal offset | vertical offset | blur | spread | color ;

So you want it on all sides means :

1. No offsets.
2. Blur as you like.

Spread here is key to this, setting 10px to the spread means 5px on all sides, basically, half the amount will be on each facing side.

* {
  margin: 0;
  padding: 0;
  box-sizing: border-box;
}

div {
  padding: 30px;
  margin: 30px;
  width: 300px;
  height: 100px;
  padding: 15px;
  background-color: orange;
  box-shadow: 0px 0px 10px 10px grey;
}

<div></div>

Also if you want to customize that you always define multiple shadows separated by a comma.

The buggy rendering in Chrome seems to be caused by the blur-radius and is hidden by the spread-radius; it seems to only occur with large blur-radius values. Through trial and error you can use the spread-radius to cover up the bug.

It's not perfect, but this works:

box-shadow: 0px 0px 140px 300px #FFF;

These are the changes that work on your site. Place the border radius and box shadow on the outer div to eliminate a gray ring.

#logo-outer {
    margin: 10px auto;
    width: 275px; /* increase width to match #logo */
    background: none repeat scroll 0% 0% #FFF;
    border-radius: 50%;
    box-shadow: 0px 0px 140px 300px #FFF; /* change the box shadow blur and spread */
}
#logo {
    height: 275px;
    width: 275px;
    background: none repeat scroll 0% 0% #FFF;
}

This is what it looks like in Chrome

Bug Workaround Example

body {
  background: #000;
}
div {
  border-radius: 50%;
  height: 250px;
  width: 250px;
  box-shadow: 0 0 140px 300px #fff;
  background: #FFF;
}

<div></div>

For circular box-shadows the blur cannot go above the width & height of the element. The spread can though.

Since your element is 200px * 200px, the maximum for the blur value is 200px.

Have a look below at the example which doesn't go above 200px and you will see that it creates the box-shadow as expected

div {
  width: 200px;
  height: 200px;
  border-radius: 100px;
  background-color: blue;
  box-shadow: 169px 129px 200px -15px rgba(0, 0, 0, 1);
}

<div></div>

The spread value can alternatively go above the element width and height and therefore you can make bigger spreads.

div {
  width: 200px;
  height: 200px;
  border-radius: 100px;
  background-color: blue;
  box-shadow: 169px 129px 0 250px rgba(0, 0, 0, 1);
}

<div></div>

You also didn't really need the prefixes since CSS3 Box-shadows are very well supported now. CanIUse

You can read more about CSS Box shadows in the MDN Documentation