The attributes that are speaking of his deity are clearly not communicable.  You mention his aseity.  I also point to his [eternal nature](http://en.wikipedia.org/wiki/Eternity#God_and_eternity). 

However, the attributes that are bound to his personality are communicalbe.  For example, [grace](http://en.wikipedia.org/wiki/Grace_(Christianity)) and [love](http://en.wikipedia.org/wiki/Love_of_God) can be [found in us](http://christianity.stackexchange.com/questions/1108/what-is-common-grace).

We have to be careful though to note that while these attributes of God's can be transferred to us, it is actually [God *in us* that brings about these attributes](http://www.biblegateway.com/passage/?search=James%201:17&version=NIV).

<sub>See also:  [Attributes of God](http://en.wikipedia.org/wiki/Attributes_of_God)</sub>