The Johnson $S_U$ distribution (see citation below) has infinite tails and can be platykurtotic based on your parameter choices. It is easily generated from normally distributed data as follows:
$$ Z = \gamma + \delta \; \text{sinh}^{-1} \left(\frac{X-\xi}{\lambda} \right) $$
where $X \sim N(0,1)$. You probably just want to set $\gamma=\xi=0$ and $\lambda=1$. The parameter $\delta$ controls kurtosis, with higher values resulting in smaller kurtosis.
Here is a contour plot (with R code) showing the bivariate density of independent $S_U$ variables:
library(SuppDists)
parms = list(gamma=0,delta=2,xi=0,lambda=2,type='SU')
coords = seq(-3,3,0.01)
zz = outer(
X=coords,Y=coords,
FUN=function(x,y) c(dJohnson(x,parms)*dJohnson(y,parms))
)
contour(coords,coords,zz)

The shape of the distribution is a bit more clear if we look at univariate densities:
curve(dJohnson(x,parms),from=-3,to=3)

Note that $\xi$ is the location parameter, meaning that it shifts the distribution left and right leaving variance, skewness, and kurtosis unchanged. $\lambda$ is the scale parameter, meaning that it changes the variance but leaves mean, skewness, and kurtosis unchanged. $\gamma$ changes skewness and $\delta$ changes kurtosis, but they affect lower moments as well.
For example, we can calculate theoretical variance and kurtosis in R when $\gamma=0,\xi=0,\delta=1,\lambda=1$:
lam1 = sJohnson(list(gamma=0,delta=1,xi=0,lambda=1,type='SU'))
lam1$Variance
##[1] 3.194528
lam1$Kurtosis
##[1] 33.18813
Note that if we change $\lambda$ to 2 and hold the other parameters constant variance changes but kurtosis remains the same:
lam2 = sJohnson(list(gamma=0,delta=1,xi=0,lambda=2,type='SU'))
lam2$Variance
##[1] 12.77811
lam2$Kurtosis
##[1] 33.18813
Note that if we then change $\delta$ to 2 and hold the other parameters constant (keep $\lambda=2$), both variance and kurtosis change:
del2lam2 = sJohnson(list(gamma=0,delta=2,xi=0,lambda=2,type='SU'))
del2lam2$Variance
##[1] 1.297443
del2lam2$Kurtosis
##[1] 1.507862
Citation: Johnson NL (1949). Systems of Frequency Curves Generated by Methods of Translation. Biometrika 36(1/2), 149-176. See in particular equation 21.
you don't want it to have a mode at zero
. $\endgroup$ – ttnphns Dec 17 '13 at 18:33