I always think it's amazing how one person's "can't live without" is someone else's total waste of space. I invested a large amount of money in a double electric breast pump and I barely use the thing. Total waste of money for me.
My most beloved purchases for my now 4.5 month old have been:
Stretchy sling - invaluable. Only way to get baby to sleep at one stage, and much easier for public transport than the pram.
Cloth wipes - cheaper and more efficient than disposable wipes.
Baby bath - lots of people say these are pointless, but I find it's much easier on my back, and she enjoys it far more than when we've bathed her in the big bath.
Co-sleeper crib - I've actually rented this from Bednest. Saves a lot of getting up and down in the night.
I'm puzzled by the posters who say they don't use muslins. I use them all the time for wiping her face, catching milk overspray when I'm feeding, to dry her bum, as an impromptu playmat...all sorts really!