If I have a Linear mixed effects model (lmer) that allows for the Intercept to vary among g1 and g2 within g1. Can g2 also be a fixed effect? I am using the lme4 package with R and would like to allow the intercept to vary with my fixed effects to allow for a proper fit, so basically I assume my random effects would follow this:
(1 | g1/g2) with the alternative  (1 | g1)+(1 | g1:g2) 
In my case my full model would be:
lmer(signal ~ 1 + g2 + g3 + (1|g1) + (1|g1:g2) + (1|g1:g3),
     data = dat, REML = FALSE)

But as you may have noticed, the variables (categorical factors g2 and g3) that I want to allow the intercept to vary for are also my fixed effects. I have not seen this in an example before and wanted to know: is it valid to do this if g2 and in my case also g3 are also fixed effects?
 A: Fixed vs Random
In this context, an effect is fixed or random. Random effects can affect fixed effects, but one explanatory variable cannot be both fixed and random.
Specifying Fixed Effects Affected by Random Effects
(1 | g1 / g2) implies g2 is a random effect nested in g1 (e.g. (1 | school / class)). It also implies you are estimating a random intercept. 

I want to allow the intercept to vary for are also my fixed effects

If you suspect the fixed effects are affected by the random effect, you should include a random slope, even for categorical variables. This may seem a little confusing, but consider the slope for your categorical variable additions to the intercept. In that case, you could model (g2 | g1) or if you believe both are affected (g2 + g3 | g1)
lme4 automatically estimates the fixed effects correctly if the random effects are correctly specified, respecting the hierarchy of the data. 
If I understand your question correctly, the model would then look something like this:  
lmer(signal ~ g2 + g3 + (g2 + g3|g1), data = dat)

You don't have to include 1 + ... in the fixed part, because an intercept is estimated by default.
Note: Unless you are comparing models, it might be better to leave REML on, because ML can be biased for the random effects, especially if the data are unbalanced.
