Is this what you are looking for? Or can you adapt this technique to get what you need?

First just show the constructed strings to try to be able to verify this is correct:

In[1]:= sumbinomial[n_] :=
 StringJoin[
  Join[
   Table["Sum[", {i, 1, n}],
   Table["Binomial[2,i" <> ToString[i] <> "],{i" <> ToString[i] <> ",1,2}]", {i, 1, n}]
   ]
  ]

In[2]:= FullForm[sumbinomial[1]]

Out[2]//FullForm= "Sum[Binomial[2,i1],{i1,1,2}]"

In[3]:= FullForm[sumbinomial[2]]

Out[3]//FullForm= "Sum[Sum[Binomial[2,i1],{i1,1,2}]Binomial[2,i2],{i2,1,2}]"

In[4]:= FullForm[sumbinomial[3]]

Out[4]//FullForm= "Sum[Sum[Sum[Binomial[2,i1],{i1,1,2}]Binomial[2,i2],{i2,1,2}]Binomial[2,i3],{i3,1,2}]"

Now the actual function, which just wraps that with ToExpression to interpret the string and carry out the calculation:

In[5]:= sumbinomial[n_] :=
 ToExpression[
  StringJoin[
   Join[
    Table["Sum[", {i, 1, n}],
    Table["Binomial[2,i" <> ToString[i] <> "],{i" <> ToString[i] <> ",1,2}]", {i, 1, n}]
    ]
   ]
  ]

In[6]:= sumbinomial[1]

Out[6]= 3

In[7]:= sumbinomial[2]

Out[7]= 9

In[8]:= sumbinomial[3]

Out[8]= 27

There should be a clear, simple, easy to understand recursive definition which does exactly what you want, but without the string hacking. Getting all the iterations correctly might be difficult to do until you think about functional programming just the right way.