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Thin Si-doped Al-rich (Al>0.85) regrown Al(Ga)N layers were deposited on AlN on Sapphire template using metal-organic chemical vapor deposition (MOCVD) techniques. The optimization of the deposition conditions such as temperature, V/III ratio, deposition rate, and Si concentration resulted in a high charge carrier concentration (>$10^{15}/cm^{3}$) in the Si-doped Al-rich Al(Ga)N films. A pulsed deposition condition was employed to achieve a controllable Al composition greater than 95% and to prevent unintended Ga incorporation in the AlGaN material deposited using the close-coupled showerhead reactor. Also, the effect of unintentional Si incorporation on free charge carrier concentration at the regrowth interface was observed by varying the thickness of the regrown Al(Ga)N layer. A maximum charge carrier concentration of $4.8\times 10^{16}/cm^3$ and $7.5\times 10^{15}/cm^3$ were achieved for Al0.97Ga0.03N and AlN films with thickness <300 nm compared to previously reported n-Al(Ga)N films with thickness $\ge$400 nm deposited using MOCVD technique.