My power failed it was if I had no battery?

Okay, I got in my car leaving work going some where's all of a sudden my dash lights went off and the rest of my power failed it was if I had no battery it was at night the car then died I had to put it to where it could be figured out I then tried to start it it gave me a rapid clicking the sound you hear when the battery goes dead I then called someone they had my car towed and put a new battery in and said it was all good now I drove not far a again my battery ran down and I lost all power and it then died I had to push it again I again turn key to start I again get the rapid clicking so I figure the alternator is bad because it was not charging the battery so I get the alternator replaced and get a new battery and I go to start it and again only rapid clicking and when I go to start it on the dash the red battery is steady on and turn key and rapid clicking so I check all the fuses and they all checked out good so then I was told it's the starter so I get that off and it tested good so it gets back on turn key and again only rapid clicking I can't figure out as to why my car won't start please help me figure it out.