The gamlss package allows you to fit generalized additive models with both lognormal and exponential distributions, and a bunch of others, with some variety in link functions and using, if you wish, semi- or non-parametric models based on penalized splines. It's got some papers published on the algorithms used and documentation and examples linked to the site I've linked to.
With GLMs, it's generally best not to think of them as "conditional mean + error" -like models but as "conditional distribution" models.
In the case of the Gamma model, note that the variance is proportional to the square of the mean. If you really want to write an error term and you have a log link, you can either write it as an additive error model on the log-scale (with constant variance) or on the original scale as a multiplicative error model (but with changing variance). I wouldn't do it as an additive model on the original scale.
The log of a gamma random variable isn't at all bad to deal with, so the additive error version is kind of convenient, if you want to deal with an error-model.
Beware, however -- the additive error version results in a term with a non-zero mean (it's also left skew, but that's less of a big deal). It's easy enough to compute an adjustment for that non-zero mean, though, so that you can correct for bias on the log-scale (or you could even fit a least-squares model to the logs and compute an adjustment for the original scale).
It seems that the "quasi-likelihood" must be applied in order to solve the models for lognormal models without log-transform.
In fact, if you want ML estimation, taking logs is basically the most sensible way to estimate the parameters of that log-normal model.
To try to do that on the original scale is just making your life hard.
See here, starting at "The MLE is also invariant with respect to certain transformations of the data." down to "For example, the MLE parameters of the log-normal distribution are the same as those of the normal distribution fitted to the logarithm of the data."
Note that lognormal and gamma models aren't the only models which would be suitable for a model that's linear in the logs, and which has constant variance on the log-scale, they just happen to both be quite convenient.
Best Answer
It depends on the nature of your dependent variable:
Gaussian is for continuous DV (this is ordinary least squares)
Binomial, as you note, is for logistic regression .
Poisson is for count data (non-negative integers). See also quasipoisson.
Gamma is for continuous DV that is always positive (although often you can use Gaussian here, if the mean is $>> 0$ and the sd isn't huge - that is, if all the values are quite far from 0).
Inverse Gaussian is, I believe, used for survival data (time to event).