Are there any training maneuvers that are only allowed when flying with a flight instructor?
Per the regs, no.  That’s the purpose of solo flying - to allow a student pilot to practice their skills on their own within a limited and safe environment.  However if you are renting the airplane, there may be prohibitions on certain maneuvers or operations eg spins, landing on unpaved runways, etc, so you will have to abide by those agreements.  Your CFI may place limitations on your solo endorsement at their discretion prohibiting certain flying for safety reasons which a student pilot must abide by.  As to any maneuvers required for a practical examination for a pilot certificate or rating, no there are no restrictions against flying them solo, provided they don’t conflict with other regulations eg operating an aircraft in controlled airspace on instruments without an instrument rating, etc.

Some schools also have limits as to what a student pilot can do solo.  I worked for a part 141 flight school once which forbade doing virtually every maneuver ie slow flight, stalls, steep turns, ground reference maneuvers, etc.  This just left the four fundamentals and changing heading, altitude and airspeed during flight.  They were training foreign students for the airlines and were pretty stringent.  I always though excessive.  But I’m sure they have plenty of bad memories where “you give ‘em an inch, they’ll take a mile”