How do I stack widgets overlapping each other in flutter

You can use a Stack with Positioned to achieve this:

class StackExample extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return new Scaffold(
      appBar: new AppBar(),
      body:  new Container(
        padding: const EdgeInsets.all(8.0),
          height: 500.0,
          width: 500.0,
          // alignment: FractionalOffset.center,
          child: new Stack(
            //alignment:new Alignment(x, y)
            children: <Widget>[
              new Icon(Icons.monetization_on, size: 36.0, color: const Color.fromRGBO(218, 165, 32, 1.0)),
              new Positioned(
                left: 20.0,
                child: new Icon(Icons.monetization_on, size: 36.0, color: const Color.fromRGBO(218, 165, 32, 1.0)),
              ),
              new Positioned(
                left:40.0,
                child: new Icon(Icons.monetization_on, size: 36.0, color: const Color.fromRGBO(218, 165, 32, 1.0)),
              )

            ],
          ),
        ),
      )
    ;
  }
}

And this how you get some nice shadow drop so the icon stands out more:

class StackExample extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return new Scaffold(
      appBar: new AppBar(),
      body:  new Container(
        padding: const EdgeInsets.all(8.0),
          height: 500.0,
          width: 500.0,
          // alignment: FractionalOffset.center,
          child: new Stack(
            //alignment:new Alignment(x, y)
            children: <Widget>[
              new Container(
                decoration: new BoxDecoration(
                  borderRadius: BorderRadius.circular(25.0),
                  boxShadow: [
                    new BoxShadow(
                      blurRadius: 5.0,
                      offset: const Offset(3.0, 0.0),
                      color: Colors.grey,
                    )
                  ]
                ),
                child: new Icon(Icons.monetization_on, size: 36.0, color: const Color.fromRGBO(218, 165, 32, 1.0))),
              new Positioned(
                left: 20.0,
                child: new Container(
                  decoration: new BoxDecoration(
                  borderRadius: BorderRadius.circular(25.0),
                  boxShadow: [
                    new BoxShadow(
                      blurRadius: 5.0,
                      offset: const Offset(3.0, 0.0),
                      color: Colors.grey,
                    )
                  ]
                ),
                  child: new Icon(Icons.monetization_on, size: 36.0, color: const Color.fromRGBO(218, 165, 32, 1.0))),
              ),
              new Positioned(
                left:40.0,
                child: new Container(
                  decoration: new BoxDecoration(
                  borderRadius: BorderRadius.circular(25.0),
                  boxShadow: [
                    new BoxShadow(
                      blurRadius: 5.0,
                      offset: const Offset(3.0, 0.0),
                      color: Colors.grey,
                    )
                  ]
                )
                  ,child: new Icon(Icons.monetization_on, size: 36.0, color: const Color.fromRGBO(218, 165, 32, 1.0))),
              )

            ],
          ),
        ),
      )
    ;
  }
}

As of November 2019 I'd like to add a second solution:

Using package: https://pub.dev/packages/assorted_layout_widgets

var widget1 = ...;
var widget2 = ...;

RowSuper(  
  children: [widget1, widget2],    
  innerDistance: -20.0,
  );

This will overlap row cells by 20 pixels.

The difference from this solution to the one using Stack is that Positioned widgets in a Stack don't occupy space. So you can't make the Stack the size of its contents, unless you know their sizes in advance. However, the RowSuper will have the size of all of its children widgets.

Note, there is also a ColumnSuper. Also note I am the author of this package.