How to overlap images in react-native

I review your code and make some changes to get your expected output. The updated code is:-

'use strict';

var React = require('react-native');                                                                                                                                                                                 
var {                                                                                                                                                                                                                
  AppRegistry,                                                                                                                                                                                                     
  StyleSheet,                                                                                                                                                                                                      
  Text,                                                                                                                                                                                                            
  View,                                                                                                                                                                                                            
} = React;                                                                                                                                                                                                           

var SampleApp = React.createClass({                                                                                                                                                                                  
  render: function() {                                                                                                                                                                                             
    return (                                                                                                                                                                                                     
      <View style={styles.container}>                                                                                                                                                                      
      <View style={styles.avatar}>                                                                                                                                                                         
      <View style={styles.badge} />                                                                                                                                                                        
      </View>                                                                                                                                                                                              
      </View>                                                                                                                                                                                              
    );                                                                                                                                                                                                           
  }                                                                                                                                                                                                                
});                                                                                                                                                                                                                  

var styles = StyleSheet.create({                                                                                                                                                                                     
  container: {                                                                                                                                                                                                     
  },                                                                                                                                                                                                               
  avatar: {                                                                                                                                                                                                        
    backgroundColor: 'black',                                                                                                                                                                                    
    width: 60,                                                                                                                                                                                                   
    height: 60,                                                                                                                                                                                                  
  },                                                                                                                                                                                                               
  badge: {                                                                                                                                                                                                         
     backgroundColor: 'red',                                                                                                                                                                                      
     width: 20,                                                                                                                                                                                                   
     height: 20,                                                                                                                                                                                                  
     left: 20,                                                                                                                                                                                                    
     top: 20,                                                                                                                                                                                                     
  },                                                                                                                                                                                                               
});                                                                                                                                                                                                                  

AppRegistry.registerComponent('SampleApp', () => SampleApp);                                                                                                                                                            

See the change in above code snippet. Output screenshot link:- https://drive.google.com/file/d/0B_8x_Jy7Ac9bbDh1eHhfelJpSmc/view?usp=sharing

Whenever you want to override any react component simply put that component in between start and close of another component. For example:-

If you want to overlap one image on another then use tags like

    <Image source={require('image!firstimage')} style={..}>
      <Image source={require('image!secondimage')} style={..}>
    </Image>

Nesting Image components doesn't work anymore. What you could use is ImageBackground instead or absolute positioning.

As doc says you can code your own specific component by checking the source code of ImageBackground https://github.com/facebook/react-native/blob/master/Libraries/Image/ImageBackground.js