"Nine" for an Exchange client is my pick. Actively developed and new features are being added all the time. Costs $10 though but you can try for 14 days for free.

For a regular Google account the normal GMail app works just fine for me too.

Lots of other apps though:

Dropbox I concur. Great for photo backup amongst other things.

I like Google Keep for simple notes and reminders (along with location based alerts)

Tasker allows lots of control of things based on events or states of the phone allowing customisation in a lot of detail.

PassWallet allows the use of Apple passbook .pbk files for boarding passes etc.

JuiceSSH for an SSH client.

RealCalc for a good scientific calculator.

Most of these I use paid versions. For that matter Google Rewards gives you free credit for some survey answers (you don't have to always answer correctly )