gain with no side effects
It's all about competing or living with noise.

A reasonable op-amp will have an equivalent input voltage noise of about 10 nV \$/\sqrt{Hz}\$ and if you implement a 1st order low pass filter having a 3 dB point of 40 Hz, the equivalent noise bandwidth will be about 63 Hz. A 2nd order filter will have an equivalent noise BW of 48 Hz as you might anticipate.

So, how much noise is the op-amp going to produce?

10 nV * \$\sqrt{63}\$ (equivalent input noise) = 0.08 uV RMS. Added to this you have to contend with 1/f noise and this, for an op-amp like the ADA4528, is only about 100 nVp-p (because it is designed that way).

So, can you live with a noise of about 0.1 uV RMS when your signal might be as low as 5 uV RMS (SNR = 34 dB)? 

You could choose a better op-amp that has lower voltage noise and improve things by 2 or 3 times. You could even adopt techniques that band limit the signal to the frequency spectrum where it exists (think I.F. strips on radio receivers).

If your signal is from a high impedance source then you have further problems due to op-amp current noises and possibly thermal noise from your source impedance. 

Lots of things to think about but if you are prepared to be realistic then you might get a decent enough solution. It's all about noise not gain. Gain is the easy thing, living with noise is the harder thing.