Problems starting 2005 Toyota Camry XLE

I am having trouble starting my Camry, and the very same thing happened about a year
ago. Back then, despite having it in the shop multiple times, and sometimes leaving it
with them for days, the car would never act up for them, and their tests showed no
problems, so nothing was ever fixed. And I kept getting stranded. Now it's "deja vu all
over again"!

Intermittently, the car will start to turn over, making all the correct sounds, except for the
actual "taking hold" sound. (There is definite noise, and it is not "clicking.") The battery is
not very old, and is testing out perfectly fine at multiple places, just like last year. And
yes, all the battery connections are clean and tight!

I have a portable plug-in starter at home, and occasionally that will start it, but more
often than not, I have to leave it plugged in for a couple of hours, then it will start
(usually). The reader on the charger starts out showing about 12.8, then goes to about
13.5 and then the car will start. Sometimes I get somewhere, get stranded, car won't
start, and I wait around hoping someone will jumpstart it (I always carry my own cables),
but sometimes no one will. Then I'll sit in frustration for a little bit, try again, and it starts!

This is a random and unpredictable situation. I may be able to start it once, then when I
go back out to use the car again later, nothing. I've gone many, many months without
this problem occurring again since last year when nothing was found and nothing was
fixed.

When I went to get the battery checked today (it was fine), the guy said a 2006 Toyota
Camry had just left and the owner had an experience virtually identical to mine. Her car
was left in the shop several times for several days; nothing happened. Car won't start,
but sometimes it will. Battery also tested fine.