Using TypeDescriptor
As mentioned in comments, Luke Foust solution does not work with anonymous types.
The cast to Dictionary does not work, I propose to construct the Dictionaries using the TypeDescriptor.GetProperties method:
public static dynamic CombineDynamics(object object1, object object2)
{
  IDictionary<string, object> dictionary1 = GetKeyValueMap(object1);
  IDictionary<string, object> dictionary2 = GetKeyValueMap(object2);
  var result = new ExpandoObject();
  var d = result as IDictionary<string, object>;
  foreach (var pair in dictionary1.Concat(dictionary2))
  {
    d[pair.Key] = pair.Value;
  }
  return result;
}
private static IDictionary<string, object> GetKeyValueMap(object values)
{
  if (values == null)
  {
    return new Dictionary<string, object>();
  }
  var map = values as IDictionary<string, object>;
  if (map == null)
  {
    return map;
  }
  map = new Dictionary<string, object>();
  foreach (PropertyDescriptor descriptor in TypeDescriptor.GetProperties(values))
  {
    map.Add(descriptor.Name, descriptor.GetValue(values));
  }
  return map;
}
It's now working with anonymous types:
var a = new {foo = "foo"};
var b = new {bar = "bar"};
var c = CombineDynamics(a, b);
string foobar = c.foo + c.bar;
Notes:
My GetKeyValueMap method is based on the RouteValueDictionary class (System.Web.Routing). I've rewritten it (using ILSpy to disassemble it) because I think that a System.Web class has nothing to do with object merging.