I keep seeing agency ads for "Hajj without the lottery". Is that actually possible?

Unless you happen to know someone in the Saudi royal family or at Türkiye's Directorate of Religious Affairs (Diyanet) — no. Under Article 6 ("Hajj quota") of the Decision on the Conduct of Hajj and Umrah Travel by the Directorate of Religious Affairs, 60% of Türkiye's Hajj quota is used by the Directorate and 40% is allocated collectively to agencies; each agency registers the pilgrims it can take within its allocated quota. By decisions of the Inter-Ministerial Hajj and Umrah Board taken before each Hajj season, pilgrims are selected by a lottery drawn from citizens who renewed their Hajj registration or pre-registered, taking into account accommodation preferences and years of pre-registration, using the coefficients and ratios set for that year.