neuraln

One of R's signatures is method dispatching where users call the same named method such as predict but internally a different variant is run such as predict.lm, predict.glm, or predict.gam depending on the model object passed into it. Therefore, calling predict on an avNNet model is not the same predict on a gam model. Similarly, just as the function changes due to the input, so does the output change.

According to MSDN documents regarding the Excel #Value! error exposed as Error 2015:

#VALUE is Excel's way of saying, "There's something wrong with the way your formula is typed. Or, there's something wrong with the cells you are referencing."

Fundamentally, without seeing actual results, Excel may not be able to interpret or translate into Excel range or VBA type the result R returns from gam model especially as you describe R raises no error.

For example, per docs, the return value of the standard predict.lm is:

predict.lm produces a vector of predictions or a matrix of predictions...

However, per docs, the return value of predict.gam is a bit more nuanced:

If type=="lpmatrix" then a matrix is returned which will give a vector of linear predictor values (minus any offest) at the supplied covariate values, when applied to the model coefficient vector. Otherwise, if se.fit is TRUE then a 2 item list is returned with items (both arrays) fit and se.fit containing predictions and associated standard error estimates, otherwise an array of predictions is returned. The dimensions of the returned arrays depends on whether type is "terms" or not: if it is then the array is 2 dimensional with each term in the linear predictor separate, otherwise the array is 1 dimensional and contains the linear predictor/predicted values (or corresponding s.e.s). The linear predictor returned termwise will not include the offset or the intercept.

Altogether, consider adjusting parameters of your predict call to render a numeric vector for easy Excel interpretation and not a matrix/array or some other higher dimension R type that Excel cannot render: