If you want to see the polynomial result before reduction, you can use the method polynomial which gives you the polynomial representing the element of your binary field, as an element of the polynomial ring. That is:
sage: X_poly = X_bf.polynomial()
sage: Y_poly = Y_bf.polynomial()
sage: X_poly * Y_poly
x^222 + x^220 + x^219 + x^218 + x^217 + x^216 + x^215 + x^214 + x^213 + x^211 + x^205 + x^204 + x^203 + x^202 + x^200 + x^199 + x^198 + x^197 + x^191 + x^188 + x^186 + x^185 + x^182 + x^179 + x^178 + x^177 + x^176 + x^175 + x^172 + x^171 + x^170 + x^169 + x^163 + x^160 + x^159 + x^154 + x^152 + x^151 + x^149 + x^146 + x^143 + x^140 + x^136 + x^132 + x^131 + x^128 + x^127 + x^126 + x^125 + x^124 + x^123 + x^120 + x^119 + x^116 + x^115 + x^113 + x^111 + x^110 + x^109 + x^106 + x^105 + x^103 + x^101 + x^97 + x^96 + x^94 + x^88 + x^85 + x^83 + x^82 + x^80 + x^75 + x^70 + x^68 + x^67 + x^66 + x^65 + x^64 + x^61 + x^58 + x^57 + x^55 + x^54 + x^46 + x^44 + x^43 + x^41 + x^35 + x^34 + x^31 + x^30 + x^27 + x^24 + x^23 + x^20 + x^18 + x^17 + x^16 + x^15 + x^11 + x^8 + x^7 + x^5 + x
And if you want the result as an integer:
sage: sum(2^i for i in _.exponents()) # _ represents the latest result
10100206405814666335316365818931450998837311752455571104712025934242
