In heavy saucepan, heat butter and water to boiling over medium high heat. Remove from heat and add flour. Stir until it forms a ball and pulls away from side of pan. Transfer to large bowl. Beat in eggs, one at a time, mixing well after each egg.
Spread in bottom of 9 x 13 ungreased cake pan.
Bake for 35 minutes. Cool competly
In large bowl, mix cream cheese and 1 cup of milk and beat until smooth; add next 3 cups of milk and blend well. Add all 3 packages of instant pudding mix and beat until thickened. Spread over cooled cake. (You may need to pat down cake if it rose too much, but that is ok!)