this reminds me, there are noise diodes designed exactly for the purpose of generating noise. One supplier is noisecom.com (they were at the top of Google search). Their NC104 model provides 3 MHz bandwidth with noise output into a 2200ohm load of 3 microV/Hz^1/2, i.e. 5 milliV RMS if the full bandwidth is usable, or 1.6 mV if filtered down to 300 kHz. If the first amplifier has 7 pF input capacitance, then, together with the 2200 Ω load, the upper cutoff frequency is 10 MHz.

--and to illustrate the problems you can get into with this "big amp" naive strategy, say you are aiming to amplify your 5 milliV by factor 200 to reach 1 volt. If you have merely around (7/200)pF = 0.035 pF of parasitic capacitance or more coupling the amp output back to the amp input, then that'd produce a bogus signal at the input comparable to the real signal, probably causing you to go into wild oscillation. That's a well known problem with high gain amps used near their bandwidth limit (at which the input is coming from maximum permitted impedance while the bogo-signal feedback gets minimum impedance). OK, you maybe can escape this fate using careful amp design and careful shielding... but why go there if you do not have to? I agree noise diodes, neons, etc (noisy things) are the way to go if you do want this approach, but to make it work you'd probably want to play it safe and operate far below the bandwidth limit.