If a polynomial with real coefficients of degree $d$ has at least $d$ coefficients equal to $1$ and has $d$ real roots, what is the maximum possible value of $d$?

(Note: The roots of the polynomial do not have to be different from each other.)
1. **Define the polynomial and its properties:**
   Given a polynomial \( P(x) \) of degree \( d \) with real coefficients, it has at least \( d \) coefficients equal to 1 and \( d \) real roots. We can write:
   \[
   P(x) = x^d + x^{d-1} + \cdots + x + 1 + a \cdot x^k
   \]
   where \( a \) is a real number and \( k \) is an integer such that \( 0 \leq k \leq d-1 \).

2. **Transform the polynomial:**
   Consider the polynomial \( P(x) \) multiplied by \( (x-1) \):
   \[
   (x-1)P(x) = (x-1)\left(x^d + x^{d-1} + \cdots + x + 1 + a \cdot x^k\right)
   \]
   Simplifying, we get:
   \[
   (x-1)P(x) = x^{d+1} - 1 + a \cdot x^k(x-1)
   \]
   Let \( R(x) = x^{d+1} - 1 + a \cdot x^k(x-1) \).

3. **Analyze the roots of \( R(x) \):**
   Since \( P(x) \) has \( d \) real roots, \( R(x) \) must have \( d+1 \) real roots (including the root at \( x=1 \) from the factor \( (x-1) \)). Let these roots be \( a_1, a_2, \ldots, a_{d+1} \).

4. **Consider the constraints on \( d \):**
   If \( d \geq 5 \), we analyze the polynomial's behavior:
   - For \( k \leq d-3 \):
     \[
     \sum a_i = 0, \quad \sum a_i \cdot a_j = 0 \implies \sum (a_i)^2 = 0
     \]
     This leads to a contradiction since the sum of squares of real numbers cannot be zero unless all \( a_i \) are zero, which is not possible.

5. **Analyze the polynomial for specific values of \( d \):**
   - For \( d = 4 \):
     Consider the polynomial:
     \[
     P(x) = x^4 + x^3 + x^2 + x + 1
     \]
     This polynomial can be factored as:
     \[
     P(x) = (x-1)^2(x^2 + 3x + 1)
     \]
     This polynomial has 4 real roots, satisfying the given conditions.

6. **Conclusion:**
   The maximum possible value of \( d \) such that the polynomial has at least \( d \) coefficients equal to 1 and \( d \) real roots is \( d = 4 \).

The final answer is \( \boxed{4} \)