Some authoritarian regimes allow at least some institutionalized limits on rulers’ power. Why does this happen? Provide examples.

An authoritarian regime is often led by a single dictator or an oligarchy. Since the leadership is a very small minority, they cannot rule without the active support of a strong inner circle and the complicity of a substantially larger group of people, including the police and the military. Thus, many authoritarian regimes use some form of institutionalized limits on power, whether real or merely for optics, in order to maintain their legitimacy. Another issue is that because many wealthy democracies insist on human rights as a precondition of trade, aid, or diplomacy, maintaining at least a facade of legitimacy is important to many regimes. Finally, companies that want the benefits of trade and foreign direct investment must show some evidence of rule of law, which constitutes a constraint on the authority of the government.
Russia (under Putin), Egypt (under Sisi), Uzbekistan, Kazakhstan, and Turkmenistan are among the recent examples of countries that stage elections while not allowing genuine democracy. Although China is authoritarian, Xi Jinping, despite the increasing centralization of power, is enforcing anti-corruption regulations and limiting the ability of politicians to be corrupt in commercial endeavors or take bribes. These limits on his own power, especially the emphasis on cracking down on corruption, solidifies his power and makes the middle classes less likely to rebel against his authority since he is offering a guarantee of lawfulness and stability, rather than the sort of caprice associated with unlimited power.

Authoritarian regimes may be structured as dictatorships run by a single strongman or junta, with a straightforward, top-down flow of power, but not all cases are so simple. Structures of government are often the result of generations of historical development, including existing constitutional and legal provisions. Governments may also contain, in practice, multiple or even competing sectors that effectively limit the power of the top leadership, although this structure does not have to resemble democratic governments with their constitutional checks and balances between executive, legislative, and judicial branches.
An interesting example to consider is the government of the Islamic Republic of Iran, formed following the Islamic revolution of 1979. The lion's share of power in Iran is held by the Supreme Leader, currently Ayatollah Ali Khamenei, who sits atop a complex of clerical institutions and also controls the Revolutionary Guard and its Quds Force. However, Iran also has a popularly elected President, although candidates for that office must be approved by the Supreme Leader. The regime squashed a civic protest movement in 2009 following the disputed presidential election of 2009.
Venezuela's government, considered by some to have authoritarian qualities, represents a contrasting example. In this case, a democratically elected and highly popular leader, Hugo Chavez, implemented policies that simultaneously broadened democratic participation in civic life and concentrated political power in the office of the President. Chavez and his successor, Nicolas Maduro, represented a movement they styled as a "Bolivarian revolution," intended to distribute the nation's wealth equitably and uplift the working class. However, their actions were intended to stymie the united political opposition, which managed to capture a legislative majority in 2015. Maduro's government reacted by trying to dissolve the assembly, bringing about an ongoing constitutional crisis.