This will be an epic post - I worked around inner city Melbourne for years, as relief and permanent staff, and studied in Melbourne so got to visit a lot of places. There are SO many good options in Melbourne, but I have to say - get in early
Central:
The Alfred
City of Melbourne Centres (Swanston St, A'Beckett)
QV
RMIT City Campus
CoAsIt (Carlton)
Lady Gowrie

East:
Richmond Creche & Kindergarten
Yarraberg Children's Centre
Glenferrie ECC
West Hawthorn ECC
Swinburne Co-op
Malin Friends
MLC Kindle
Belford Oaks
Through Rd
Wattletree ECC
Wattletree House
Minifie Park
Greythorn ECC
Deakin Childcare Co-op
Wattle Park CSC
Central Box Hill CSC
Watts St
Lucknow St
Hansen St
Best Chance
Brandon Park CC
Highgate ECC
May Armstrong
Ivanhoe Chn's Co-op
Body SHop Chn's Centre
Columbia PArk
Bentleigh Family Co-op

North:
Little Sophia
Melbourne Uni ELC
Gold Street
Annie Todd
John St
St Vincent's ELC
North Fitzroy Co-op
Lady Huntingfield
Wimble St
Nara Community
Merri Community
North Carlton Chn's Centre

West:
Kensington Community
Curzon St
Hotham Hub
Church St (West Footscray)
Dawson St
Moreland Co-op
Park St

South:
Ada Mary A'Beckett
Clark St
Lancaster House
Swinburne Prahran
Princes Close
Coventry CCC
Emerald Hill
Fawkner Park
Eildon Rd
St Kilda North Chn's Centre
St Kilda Chn's Centre
St Kilda - Balaclava Kindergarten
Tree House St K/ Elsternwick
King David
Hornbrook
Albert Park College
South Melbourne Co-op
Hunt Community
Elsternwick Chn's Centre
Sholem Aleichem
Poets Grove
Elwood Children's ctr

That's what I can come up with for now. I'm sure there's more though!