The largest prime with no repeated digits is

987654103

It couldn't have 10 digits, as that would be a number divisible by 9, so a digit not divisible by 3 had to be removed to make it indivisible by 3. To give it the maximal number of remaining digits it needed a zero to give it nine full digits, but not at either the end nor the beginning.

The answer does indeed lack the digit 2, and has its zero in the second to last position. All the candidates were tested by 

DefDbl A-Z

Dim crlf$

Private Sub Form_Load()

 Text1.Text = ""

 crlf$ = Chr(13) + Chr(10)

 

 s$ = "123456789": h$ = s

 Do

   If Val(Right(s, 1)) Mod 3 <> 0 Then

     s1$ = Left(s, 8)

     For p = 1 To 7

       s2$ = Left(s1, p) + "0" + Mid(s1, p + 1)

       v = Val(s2)

       pd = prmdiv(v)

       If pd = v Then

          If v > mx Then mx = v

       End If

     Next

   End If

   DoEvents

   permute s

 Loop Until s = h

 

 Text1.Text = mx

 

 Form1.Visible = True

End Sub

 

Function prmdiv(num)

 Dim n, dv, q

 If num = 1 Then prmdiv = 1: Exit Function

 n = Abs(num): If n > 0 Then limit = Sqr(n) Else limit = 0

 If limit <> Int(limit) Then limit = Int(limit + 1)

 dv = 2: GoSub DivideIt

 dv = 3: GoSub DivideIt

 dv = 5: GoSub DivideIt

 dv = 7

 Do Until dv > limit

   GoSub DivideIt: dv = dv + 4 '11

   GoSub DivideIt: dv = dv + 2 '13

   GoSub DivideIt: dv = dv + 4 '17

   GoSub DivideIt: dv = dv + 2 '19

   GoSub DivideIt: dv = dv + 4 '23

   GoSub DivideIt: dv = dv + 6 '29

   GoSub DivideIt: dv = dv + 2 '31

   GoSub DivideIt: dv = dv + 6 '37

 Loop

 If n > 1 Then prmdiv = n

 Exit Function

DivideIt:

 Do

  q = Int(n / dv)

  If q * dv = n And n > 0 Then

    prmdiv = dv: Exit Function

   Else

    Exit Do

  End If

 Loop

 Return

End Function