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1. What happens when the I2C pullups are omitted?

There will be no communication on the I²C bus. At all. The MCU will not be able to generate the I²C start condition. The MCU will not be able to transmit the I²C address.

Wondering why it worked for 3 months? Read on.

2. The lack of pullups is likely to damage any of those two ICs in my board?

Probably not. In this particular case (MCU, RTC, nothing else), definitely not.

3. Why was the MCU able to communicate with the I²C slave device in the first place? I²C requires pull-up resistors. But they weren't included in the schematic.

Probably, you have internal pull-ups on the enabled on the ATmega. From what I've read¹, ATmega have 20kΩ internal pull-ups, which can be enabled or disabled from the firmware. 20kΩ is way too weak for the I²C pull-up. But if the bus has a low capacitance (physically small) and communication is slow enough, then 20kΩ can still make the bus work. However, this is not a good reliable design, compared to using discrete pull-up resistors.

¹Not an ATmega guy myself.

update: In response I²C waveforms, which were added to the O.P.
The waveforms in the O.P. have a very long raise time constant. Here's what I²C waveform usually looks like

PIC18F4550, Vcc=+5V, 2.2kΩ pull ups. Waveform shows SCL. The raise time on SDA is about the same. The physical size of the bus is moderate: 2 slave devices, PCB length ≈100mm.

1. What happens when the I2C pullups are omitted?

There will be no communication on the I²C bus. At all. The MCU will not be able to generate the I²C start condition. The MCU will not be able to transmit the I²C address.

Wondering why it worked for 3 months? Read on.

2. The lack of pullups is likely to damage any of those two ICs in my board?

Probably not. In this particular case (MCU, RTC, nothing else), definitely not.

3. Why was the MCU able to communicate with the I²C slave device in the first place? I²C requires pull-up resistors. But they weren't included in the schematic.

Probably, you have internal pull-ups enabled on the ATmega. From what I've read¹, ATmega have 20kΩ internal pull-ups, which can be enabled or disabled from the firmware. 20kΩ is way too weak for the I²C pull-up. But if the bus has a low capacitance (physically small) and communication is slow enough, then 20kΩ can still make the bus work. However, this is not a good reliable design, compared to using discrete pull-up resistors.

¹Not an ATmega guy myself.

update: In response I²C waveforms, which were added to the O.P.
The waveforms in the O.P. have a very long rise time constant. Here's what I²C waveforms usually look like

PIC18F4550, Vcc=+5V, 2.2kΩ pull ups. Waveform shows SCL. The rise time on SDA is about the same. The physical size of the bus is moderate: 2 slave devices, PCB length ≈100mm.

1. What happens when the I2C pullups are omitted?

There will be no communication on the I²C bus. At all. The MCU will not be able to generate the I²C start condition. The MCU will not be able to transmit the I²C address.

Wondering why it worked for 3 months? Read on.

2. The lack of pullups is likely to damage any of those two ICs in my board?

Probably not. In this particular case (MCU, RTC, nothing else), definitely not.

3. Why was the MCU able to communicate with the I²C slave device in the first place? I²C requires pull-up resistors. But they weren't included in the schematic.

Probably, you have internal pull-ups on the enabled on the ATmega. From what I've read¹, ATmega have 20kΩ internal pull-ups, which can be enabled or disabled from the firmware. 20kΩ is way too weak for the I²C pull-up. But if the bus has a low capacitance (physically small) and communication is slow enough, then 20kΩ can still make the bus work. However, this is not a good reliable design, compared to using discrete pull-up resistors.

¹Not an ATmega guy myself.

1. What happens when the I2C pullups are omitted?

There will be no communication on the I²C bus. At all. The MCU will not be able to generate the I²C start condition. The MCU will not be able to transmit the I²C address.

Wondering why it worked for 3 months? Read on.

2. The lack of pullups is likely to damage any of those two ICs in my board?

Probably not. In this particular case (MCU, RTC, nothing else), definitely not.

3. Why was the MCU able to communicate with the I²C slave device in the first place? I²C requires pull-up resistors. But they weren't included in the schematic.

Probably, you have internal pull-ups on the enabled on the ATmega. From what I've read¹, ATmega have 20kΩ internal pull-ups, which can be enabled or disabled from the firmware. 20kΩ is way too weak for the I²C pull-up. But if the bus has a low capacitance (physically small) and communication is slow enough, then 20kΩ can still make the bus work. However, this is not a good reliable design, compared to using discrete pull-up resistors.

¹Not an ATmega guy myself.

update: In response I²C waveforms, which were added to the O.P.
The waveforms in the O.P. have a very long raise time constant. Here's what I²C waveform usually looks like

PIC18F4550, Vcc=+5V, 2.2kΩ pull ups. Waveform shows SCL. The raise time on SDA is about the same. The physical size of the bus is moderate: 2 slave devices, PCB length ≈100mm.

1. Q: What happens when the I2C pullups are omitted?
   A: There will be no communication on the I2C bus. At all. The μC will not be able to generate the I²C start condition. The μC will not be able to transmit the I²C address.
   [Wondering why it worked for 3 moths? Read on.]

2. Q: The lack of pullups is likely to damage any of those two ICs in my board?
   A: Probably not. In this particular case (μC, RTC, nothing else), definitely not.

3. Q: Why was the μC able to communicate with the I²C slave device in the first place? I²C requires pull-up resistors. But they weren't included in the schematic.
   A: Probably, you have internal pull-ups on the enabled on the ATmega. From what I've read**, ATmega have 20kΩ internal pull-ups, which can be enabled or disabled from the firmware. 20kΩ is way too weak for the I²C pull-up. But if the bus has a low capacitance (physically small) and communication is slow enough, then 20kΩ can still make the bus work. However, this is not a good reliable design, compared to using discrete pull-up resistors.

** not an ATmega guy myself

1. What happens when the I2C pullups are omitted?

There will be no communication on the I²C bus. At all. The MCU will not be able to generate the I²C start condition. The MCU will not be able to transmit the I²C address.

Wondering why it worked for 3 months? Read on.

2. The lack of pullups is likely to damage any of those two ICs in my board?

Probably not. In this particular case (MCU, RTC, nothing else), definitely not.

3. Why was the MCU able to communicate with the I²C slave device in the first place? I²C requires pull-up resistors. But they weren't included in the schematic.

Probably, you have internal pull-ups on the enabled on the ATmega. From what I've read¹, ATmega have 20kΩ internal pull-ups, which can be enabled or disabled from the firmware. 20kΩ is way too weak for the I²C pull-up. But if the bus has a low capacitance (physically small) and communication is slow enough, then 20kΩ can still make the bus work. However, this is not a good reliable design, compared to using discrete pull-up resistors.

¹Not an ATmega guy myself.