You are replacing the starting tag and then putting that back in innerHTML, so the code will be invalid. Make all the replacements before you put the code back in the element:
var html = strMessage1.innerHTML;
html = html.replace( /aaaaaa./g,'<a href=\"http://www.google.com/');
html = html.replace( /.bbbbbb/g,'/world\">Helloworld</a>');
strMessage1.innerHTML = html;
You should chain the replace() together instead of assigning the result and replacing again.
var strMessage1 = document.getElementById("element1") ;
strMessage1.innerHTML = strMessage1.innerHTML
.replace(/aaaaaa./g,'<a href=\"http://www.google.com/')
.replace(/.bbbbbb/g,'/world\">Helloworld</a>');
See DEMO.