One Unif[0,1] random variable:

f[x_]=PDF[UniformDistribution[],x]

Plot[f[x],{x,-1,2}]

Sum of two Unif[0,1] random variables:

f2[x_]=Convolve[f[t],f[t],t,x]

Plot[f2[x],{x,-1,3}]

Sum of three Unif[0,1] random variables:

f3[x_]=Convolve[f[t],f2[t],t,x]

Plot[{f3[x],PDF[NormalDistribution[3/2,σ Sqrt[3]],x]},{x,-1,4}]

σ=StandardDeviation[UniformDistribution[]]

Sum of four Unif[0,1] random variables:

f4[x_]=Convolve[f[t],f3[t],t,x]

Plot[{f4[x],PDF[NormalDistribution[4/2,σ Sqrt[4]],x]},{x,-1,5}]

Sum of eight Unif[0,1] random variables:

f8[x_]=Convolve[f4[t],f4[t],t,x]

Plot[{f8[x],PDF[NormalDistribution[8/2,σ Sqrt[8]],x]},{x,-1,9}]