1.js

class List extends Component {
  constructor(props) {
    super(props);
    
    this.state = {
      list: [1, 2, 3, 4, 5]
    };
  }
  
  render() {
    return (
      <div>
        <ul>
          {this.state.list.map(num => {
            return <li key={num}>{num}</li>
          })}
        </ul>
        <button onClick={this.addElement.bind(this)}>add number</button>
      </div>
    );
  }
  
  addElement() {
    this.setState({ // полный перерендер компонента
      list: this.state.list.concat(this.state.list.length + 1)
    });
  }
}