I find that my necessities change every few weeks or so, as LO gets older he likes things he didn't seem to notice before and he gets tired of others.
Swaddle sack when he was a newborn. Worked like a dream.
Long-sleeve onesies with the built-in "mittens." The baby mittens just got pulled off too easily.
Diaper caddy is nice to keep that area organized.
Large changing mat for the diaper bag.
Boppy pillow - not so much for breastfeeding as for LO to lounge in, now I use it to support him when he's sitting up.
His rocker with mobile overhead was great for a while so I could get a little housework done. He doesn't seem to tolerate it as much now.
Musical toys.
Pacifiers.
Bottle warmer! I can't imagine having to boil water all the time.
That brings me to our microwave and the microwave sterilizers - bags or trays. Boiling water every day in the summer was no bueno. We were happy to not have a microwave when it was just the two of us. But I absolutely couldn't live without it now that LO is around.
A few things that proved unnecessary:
Side-snap t-shirts for newborns. These were suggested for the healing umbilical cord. They ride up and get all twisted, plus light cotton onesies really will not create too much friction against that area.
Receiving blankets. i got way too many small flannel receiving blankets that were barely big enough to swaddle my newborn. I wouldn't think you need any more than 2 or 3, personally. Bigger is better for us.
Mittens - just get pulled off and lost too easily.
Baby towels? I don't know why I thought our towels wouldn't be just fine. The baby towels seem kinda scratchy too.
Baby bath. Not for us. Bathed him in the sink first 5 months. Now trying the big tub. It's a little tricky, but getting easier.